•     •  ••  .   .:.♦:••• 


TEACnEllS'  MANUAL"-^"  • 


FOR  TEACHERS  USING 


ARITHMETIC    BY    GRADES 


BY 


JOHN  T.  PRINCE,  Ph.D. 


AUTHOR  OF  "COURSES  OF  STUDIES  AND  METHODS  OF  TEACHING,"  "METHODS  OF 
INSTRUCTION  AND  ORGANIZATION  OF  THE  SCHOOLS  OF  GERMANY,"  ETC. 


BOSTON,  U.S.A. 
PUBLISHED    I;Y   GINN   &    C().MPANY 

1895 


Copyright,  1894 
By  JOKN  T.  PKINCE 


ALL  RIGHTS  RESERVED 


PREFACE. 


Considering  the  amount  of  time  given  to  the  subject  of  Arith- 
metic in  our  schools,  it  must  be  admitted  that  the  results  are 
lamentably  poor.  It  is  fair  to  presume  that  the  faulty  character 
of  the  text-books  in  general  use  is  accountable  in  part  for  these 
results.  With  few  exceptions  American  text-books  in  Arithmetic 
seem  to  leave  the  teacher  out  of  the  account.  Definitions,  rules, 
explanations,  illustrative  problems  —  in  short,  everything  which 
should  be  taught  by  the  teacher  is  placed  before  the  pupil  to  be 
learned.  It  is  true  that  the  best  teachers  generally  ignore  these 
false  aids  and  ask  the  pupils  not  to  refer  to  them.  Yet,  because 
they  are  constantly  before  the  pupils,  the  forbidden  aids  are  often- 
times u.nwisely  used,  and  problems  are  performed  slyly  or  thought- 
lessly according  to  the  rule  or  model  solution.  While  these  practices 
are  carried  on  to  some  extent  in  good  schools,  despite  the  care  of 
teachers,  they  are  universally  pursued  in  schools  whose  teachers 
are  guided  by  the  text-book.  Another  fault  of- text-books  in  com- 
mon use  is  the  insufficiency  of  problems  for  practice,  and  teachers 
are  compelled  to  make  up  the  deficiency  by  placing  upon  the  black- 
board original  problems  or  problems  taken  from  Arithmetics  other 
than  the  regular  text-book.  The  objections  to  this  practice  are  : 
(1)  The  danger  for  want  of  time  of  giving  to  the  pupils  unsuital)le 
or  poorly-arranged  problems  ;  (2)  the  harm  done  to  pupils'  eyes 
by  close  and  long-continued  looking,  frequently  before  a  lighted 
Aviiidow;   (J))   a-  loss  of  tlie  teacher's  time  in  copying. 

The  series  of  books  of  which  this  Manuvd  is  a  juirt  is  designed 
to  meet  the  above  objections.  It  will  be  readily  seen  tluit  the 
pupils'  books  are  not  merely  books  of  problems  intended  to  supple- 
ment the  use  of  an  arithmetic  already  in  the  hands  of  pupils,  but 

54  I /36 


IV  PEEFACE. 

are  intended  to  be  used  independently  as  the  only  text-books  needed. 
Moreover,  the  very  large  number  and  variety  of  problems  Avarrant 
the  assertion  that  the  books  contain  all  the  problems  that  will  be 
needed  for  drill  work. 

The  books  for  pupils  are  eight  in  number,  arranged  somewhat 
on  the  lines  of  classification  in  City  graded  schools.  The  first  two 
or  three  books  are  intended  for  use  in  Primary  schools,  the  last 
book  in  advanced  Grammar  schools  or  High  schools.  The  subjects 
are  divided  as  follows : 

Book  I. :  Numbers  from  1  to  20. 

Book  II. :  Numbers  from  1  to  100. 

Booh  III :  Integers  to  1000000,  Fractional  Parts  of  Numbers, 
U.  S.  Money,  Common  Weights  and  Measures. 

Book  IV. :  Whole  Numbers  unlimited,  Common  Fractions  to 
Twelfths,  Decimal  Fractions  to  Thousandths,  Measurements,  Busi- 
ness Transactions,  Denominate  Numbers. 

Book  V. :  Common  and  Decimal  Fractions,  Mensuration,  Denomi- 
nate Numbers,  Business  Transactions. 

Book  VI. :  Mensuration,  Denominate  Numbers,  Metric  System, 
Percentage  and  Simple  Applications,  Business  Transactions  and 
Accounts. 

Book  VII. :  Profit  and  Loss,  Commission,  Insurance,  Taxes, 
Duties,  Interest,  Banking,  Stocks  and  Bonds,  Exchange,  Business 
Accounts,  Geometrical  Exercises  and  Measurements,  Eatio  and 
Proportion. 

Book  VIII. :  Miscellaneous  questions  involving  the  making  of 
definitions,  rules  and  formulas  ;  Algebraic  Exercises,  Involution 
iiiul  Iv.(iluiii)ii,  Exercises  in  Geometry  and  Mensuration,  Book- 
keeping. 

It  has  been  tlie  aim  of  the  author  to  include  in  the  books  all 
subjects  that  are  likely  to   be  needed  in  any  school,  rather  than 


PREFACE.  V 

limit  them  to  the  possible  needs  of  a  certain  class  of  schools.  A 
selection  of  subjects,  therefore,  as  well  as  a  selection  of  problems 
under  a  given  subject  may  be  made  to  suit  existing  conditions. 

The  following  advantages  are  claimed  for  the  use  of  the  Manual 
and  text-books  : 

1.  The  separation  of  teachers'  and  pupils'  books,  whereby  pupils 
may  be  taught  properly  and  may  not  be  given  too  great  assistance. 
Suggestions  as  to  methods  of  teaching  and  drilling,  as  well  as  the 
illustrative  processes,  explanations,  rules,  and  definitions  which 
belong  to  the  teacher  to  develop  analytically  are  put  into  the 
Teachers'  Manual,  while  in  the  pupils'  books  are  presented  only 
such  exercises  as  are  needed  for  practice. 

2.  The  careful  gradation  of  problems,  by  which  pupils  acquire 
inductively  a  knowledge  of  arithmetical  relations  and  principles, 
and  skill  in  arithmetical  processes.  This  is  in  recognition  of  the 
well-known  pedagogical  principles  of  proceeding  from  the  known 
to  the  unknown,  and  from  the  simple  to  the  complex.  It  is  advised 
that  this  plan  be  kept  constantly  in  mind  by  the  teacher,  and  that 
whenever  a  process  is  not  understood  or  is  not  readily  performed, 
the  pupils  be  taken  back  to  processes  which  are  well  known  and 
which  can  be  performed  readily,  and  then  be  led  forward  by  easy 
steps  until  the  desired  end  is  reached. 

3.  Frequent  reviews,  and  such  an  arrangement  of  exercises  as 
will  enable  pupils  to  have  needed  practice  in  the  applications  of 
each  principle,  first  by  itself,  and  afterwards  in  connection  with 
other  principles  which  have  been  learned. 

4.  The  large  amount  of  oral  work,  or  work  which  may  be  done 
without  tlie  aid  of  figures.  Three  objects  of  Mental  Arithmetic 
are  sought  in  these  exercises  :  (a)  Illustration  of  principles  and  a 
preparation  for  written  work,  (b)  Development  of  the  logical 
powers,  (c)  Cultivation  of  ability  to  work  with  large  numbers  by 
short  processes. 

5.  The  great  number  and  variety  of  problems.  The  aim  has 
been  to  give  the  largest  number  of  problems  that  will  be  needed 


VI  PREFACE. 

for  teaching  and  for  drilling  in  all  grades.  For  this  reason,  and 
because  the  forms  of  expression  are  varied,  being  taken  from  many 
sources,  there  will  be  no  necessity  of  giving  supplementary  drill 
lessons  on  the  blackboard. 

6.  Practicalness  of  work  in  respect  to  the  character  of  the  jjrob- 
lems,  and  the  solution  of  them.  Care  has  been  taken  to  give 
problems  which  are  most  likely  to  be  met  in  every-day  life,  and  to 
give  tliem  in  a  practical  form.  Many  of  the  miscellaneous  review 
problems  were  made  by  mechanics,  clerks,  accountants,  etc.,  with  a 
view  of  presenting  conditions  most  likely  to  occur. 

7.  The  introduction  of  statistics  and  facts  of  physics,  astronomy, 
history,  geography,  etc.,  thus  enabling  pupils  to  gain  incidentally 
much  useful  information. 

8.  The  use  of  drill  tables  and  other  devices  to  save  the  time  of 
teachers. 

In  addition  to  the  above  features,  some  of  which  are  distinctively 
new  so  far  as  American  text-books  are  concerned,  there  is  the 
separation  of  pupils'  exercises  for  practice  into  small  books  some- 
what on  the  lines  of  gradation  in  City  graded  schools.  By  this  ar- 
rangement there  are  gained  greater  convenience  of  handling  and 
economy  of  wear  than  in  the  use  of  a  large  book  which  is  intended 
to  be  used  for  several  years  by  the  same  pupil. 

The  author  is  aware  that  the  use  of  tliese  books  with  accom- 
panying manual  in  the  manner  proposed  is  not  an  experiment. 
Essentially  the  same  plan  has  been  pursued  in  Germany  for  many 
years,  and  it  is  confidently  believed  that  American  teachers  will 
readily  recognize  its  merits. 

The  work  of  bringing  together  so  large  a  number  of  exercises  has 
been  materially  lessened  by  the  valuable  contributions  of  tea'chers, 
business  men,  and  mechanics,  who  generously  responded  to  the 
author's  request  for  assistance.  Especial  acknowledgments  for  such 
service  should  be  made  to  Misses  Smith,  Dale,  and  Barber  of  the 
Practice  Department  connected  with  the  Framingham,  Mass.,  Normal 
School ;  Miss  A.  Roof  and  Mr.  B.  W.  Drake,  of  Waltham,  Mass.; 


PREFACE.  VU 

and  Mr.  L.  F.  Warren,  of  West  Newton,  Mass.  Obligations  for 
valuable  assistance  in  correcting  proof  sheets  are  also  due  to  Miss 
F.  A.  Comstock,  of  the  Bridgewater  Normal  School,  and  Miss  Amelia 
Davis,  of  the  Framingham  Normal  School. 

Suggestions  for  improving  the  work  and  corrections  of  errors, 
either  in  the  INIanual  or  Pupils'  Books,  will  be  gratefully  received 
by  the  author. 


CONTENTS. 


SECTIOX 

I.    General  Suggestions 


Ends  —  Course  of  Study  —  The  Recitation  —  Methods  of 
Drill  —  Reviews  —  Tasks  and  Desk- Work  —  Illustrations — 
Reading  Problems  —  Analyses  and  Explanations  —  Trob- 
lems  and  Indicated  Operations  —  Eorecasting  Answers  — 
Language. 


II.    First  Steps  in  Number 


Grouping  of  Pupils  —  Counters  —  Methods  —  Pacts  of  Num- 
bers to  Ten  —  Picturing  of  Problems  —  Use  of  Figures  — 
Number  of  Stories. 


III.  Notes  for  Book  Number  One 

IV.  Notes  for  Book  Number  Two 
V.  Notes  for  Book  Nnmber  Three 

VI.  Notes  for  Book  Number  Four 

VII.  Notes  for  Book  Numlier  Five 

VIII.  Notes  for  Book  Number  Six 

IX.  Notes  for  Book  Number  Seven 


X.    Notes  for  Book  Number  Eight 


PAGES 

1-8 


8-13 


13-21 

21-34 

35-5G 

56-81 

82-1 1 1 

111-139 

139-178 

179-225 


,'  > » ' ; 


SECTION  I. 

GENERAL  SUGGESTIONS. 

Ends.-— Arithmetic  like  most  subjects  of  instruction  may  be 
said  to  have  two  values  —  its  practical  value  as  an  aid  in  carrying 
on  the  ordinary  affairs  of  life,  and  the  value  it  has  as  a  means  of 
mental  discipline.  The  ends  of  teaching  Arithmetic  therefore  are 
knowledge  and  power.  It  is  well  for  the  teacher  to  know  at  the 
outset  the  relative  importance  of  these  ends  for  the  purpose  of 
making  a  wise  selection  of  subjects  and  of  knowing  how  they 
should  be  taught.  If  in  our  teaching  of  Arithmetic  we  keep  only 
in  mind  the  direct  use  that  it  has  as  a  preparation  for  business,  we 
sliall  find  that  a  small  fraction  of  the  time  usually  given  to  this 
branch  of  instruction  is  sufficient.  If,  on  the  other  hand,  we  con- 
sider the  possibilities  of  a  proper  presentation  of  the  various  topics 
of  Arithmetic  in  the  cultivation  of  the  powers  of  observation, 
imagination,  reflection,  and  reasoning,  and  note  the  rare  oppor- 
tunity which  the  recitation  in  this  branch  has  for  teaching  accuracy 
and  clearness  of  expression,  we  shall  be  inclined  to  place  the  dis- 
ciplinary side  of  the  study  far  above  that  of  the  direct  assistance  it 
has  in  earning  a  living  or  in  performing  the  common  duties  of  life, 
and  give  to  Arithmetic  a  prominent  place  in  the  Course. 

Another  end  of  arithmetical  instruction  which  perhaps  is  in- 
volved in  the  ends  already  named,  is  skill  or  facility  in  the 
manipulation  of  numbers.  The  practical  as  well  as  the  disciplinary 
value  of  the  study  is  greatly  enhanced  by  such  facility  as  will  en- 
able the  pupils  to  perform  accurately  and  quickly  the  computations 
involved  in  the  problems.  It  is  the  purpose,  therefore,  of  good 
teachers  of  Arithmetic  to  lead  the  pupils  to  become  accurate  and 
quick  computers,  and  to  secure  this  end  there  is  needed  much 
practice   in   the   application   of   principles   already  known.     The 


•?f'2  :    :  GRADED   ARITKMETIC. 

puipjls  ciife  add  columns,  ascertain  measurements,  and  cast  interest 
accurately  and  quickly  only  by  the  repeated  performance  of  the 
same  mental  act,  and  for  all  such  work  abundant  practice  is 
necessary. 

Course  of  Study.  —  Although  the  comparative  unimportance  of 
the  practical  side  of  Arithmetic  may  be  recognized,  the  mistake 
should  not  be  made  of  spending  much  of  the  time  given  to  the 
study  upon  unpractical  subjects.  The  constant  aim  of  the  teacher 
should  be  to  emphasize  essentials,  and  by  essentials  is  meant  those 
subjects  and  processes  which  are  in  some  degree  a  preparation  for 
life  or  for  further  study.  There  is  abundant  opportunity  for  useful 
disciplinary  practice  in  some  of  the  common  and  simpler  subjects, 
such  as  denominate  numbers,  fractions,  and  the  applications  of  per- 
centage that  are  made  in  business. 

To  rightly  judge  what  the  essentials  of  Arithmetic  are,  a  con- 
sideration of  interests  other  than  those  of  direct  utility  is  necessary. 
The  age,  capacity,  and,  so  far  as  circumstances  permit,  the  probable 
future  occupation  of  the  pupils  should  be  considered.  Tavo  kinds 
of  practice  with  numbers  are  required  of  pupils  from  the  beginning 
of  the  course  :  one  kind  which  has  to  do  with  the  mechanical  side 
of  problems  —  the  computations  with  what  are  sometimes  called 
abstract  numbers  ;  and  another  kind  which  has  to  do  with  the 
applications  and  use  of  principles,  and  which  involve  some  logical 
processes.  The  amount  of  the  latter  work  is  relatively  small  in 
the  lower  grades,  and  gradually  increases  in  amount  and  complexity 
as  tlie  pupil  grows  in  maturity,  while  the  amount  of  simple  com- 
putations for  exactness  in  the  four  fundamental  rules  gradually 
decreases  until,  in  the  latter  part  of  the  course,  nothing  but  con- 
crete or  applied  work  is  done. 

So  far  as  this  series  of  books  is  concerned,  it  is  not  siipposed 
that  all  schools  will  follow  the  exact  order  prescribed,  or  tliat  the 
books  will  exactly  correspond  with  grades  of  pupils  in  all  places. 
In  the  amount  of  work  to  be  given  also  there  will  be  a  great 
variety  of  practice  depending  upon  the  age  of  admission  of  pupils, 
the  requirements  in  otlier  departments  of  study,  and  the  intentions 


teachers'  manual. 


of  pupils  as  to  the  probable  extent  of  their  attendance  at  school. 
While  in  no  one  school  or  system  of  schools  will  it  be  needful  to 
take  up  every  subject  presented,  or  to  give  to  pupils  all  the  exercises 
of  certain  subjects,  it  is  hoped  that  both  subjects  and  exercises  are 
sufficient  in  number  and  variety  to  meet  the  needs  of  all,  and  that 
from  them  a  wise  selection  may  be  made. 

The  Recitation.  —  One  prime  purpose  of  the  recitation  is  to 
develop  the  power  of  self-direction  in  the  pupils,  —  to  lead  them 
to  depend  upon  themselves  not  only  in  accuracy  of  computations 
but  also  in  all  reasoning  processes.  For  the  purpose  of  encouraging 
pupils  to  proper  effort  in  study,  and  of  seeing  that  certain  prin- 
ciples or  processes  are  understood,  the  teacher  finds  it  necessary  to 
examine  their  completed  tasks,  or  to  test  them  upon  what  they 
have  done.  But  to  accomplish  the  objects  named,  it  is  not  neces- 
sary or  well  to  spend  the  entire  recitation  time  in  examination,  A 
glance  at  the  work  performed,  and  a  comparison  of  the  pupils' 
answers  with  the  correct  answers,  are  generally  sufficient  to  in- 
sure faithful  effort  in  study.  To  give  the  lessons  learned  such 
prominence  as  to  make  them  alone  determine  the  pupils'  promotion 
or  standing  in  the  class,  is  to  put  temptation  in  their  way  to  copy 
solutions  or  answers,  or  to  seek  assistance  that  will  not  be  helpful. 
Moreover,  the  detailed  explanations  and  solutions  of  problems 
already  performed,  frequently  mean  a  tedious  recitation  and  much 
needless  waiting  on  the  part  of  some  members  of  the  class.  If  it 
is  desired  to  ascertain  whether  the  pupils  understand  thoroughly  a 
principle  or  process  which  has  been  taught,  a  few  problems  which 
involve  the  principle  or  process  may  be  given  the  class  as  a  test. 
These  problems  should  have  small  numbers,  so  that  the  difficulties 
of  computation  Avill  not  distract  attention,  and  that  the  pupils' 
understanding  of  the  question  may  appear  in  the  answers  they  give. 
For  example,  if  the  subject  of  Profit  and  Loss  has  been  taught,  and 
the  pupils  have  broTight  to  the  recitation  the  problems  which  they 
have  solved  in  two  of  the  cases  of  that  subject,  the  teacher,  after 
glancing  at  the  work  and  leading  them  to  correct  errors  in  their 
answers,  might  give  as  a  test  the  following  problems  : 


4  GEADED   ARITHMETIC. 

A  horse  whicli  cost  $250  is  sold  for  what  to  gain  20  <^  ? 

I  buy  a  house  for  $1000  and  sell  it  immediately  for  $750.  What 
was  the  gain  or  loss  per  cent?  Suppose  I  had  sold  it  for  $1200. 
What  would  have  been  the  gain  or  loss  per  cent  ? 

From  answers  to  these  and  similar  questions  it  will  be  known 
whether  tlie  subject  is  understood,  and  time  will  be  left  for  other 
purposes  of  the  recitation. 

When  any  subject  or  process  is  not  understood,  it  should  be 
taught,  and  by  teaching  is  meant,  leading  the  pupils  to  acquire 
knowledge  by  their  own  efforts.  Little  should  be  told  the  pupils 
in  this  process.  The  teacher's  part  is  first  to  lead  the  pupils  to 
recall  ideas  already  in  the  mind  which  have  some  relation  to  the 
new  principle  or  process  to  be  taught.  Then  follows  the  presenta- 
tion of  the  object  of  thought  in  such  a  way  as  to  awaken  new  ideas 
in  the  learner's  mind,  or  to  readjust  and  reinforce  old  ones.  The 
object  of  thought  may  be  a  physical  object,  as  a  meter-stick  or  a 
bond,  or  what  represents  an  object,  as  a  picture  or  diagram. 
Sometimes,  as  in  teaching  a  rule  or  definition,  the  teacher  has 
only  to  help  the  pupils  to  recall  and  rearrange  their  ideas  in  an 
orderly  way,  and  to  make  a  general  statement  from  them.  Some 
processes  also  may  be  taught  by  leading  the  pupils  to  recall  what 
they  have  previously  learned,  and  make  the  needful  application. 
Such  purposes  may  be  accomplished  frequently  by  skilful  ques- 
tioning. 

When  the  matter  in  hand  is  taught,  and  ideas  concerning  it  are 
clear  in  the  pupils'  minds,  they  need  to  be  fixed  by  much  repeti- 
tion. This  is  done  in  the  recitation  by  drill  and  in  the  practice 
which  the  pupils  get  iu  study.  Little  time  is  needed  for  teaching 
compared  to  the  time  that  is  needed  for  practice.  A  portion  of 
nearly  every  recitation  may  well  be  given  to  drill,  both  upon  what 
has  been  taught  recently  and  ujjon  what  has  been  taught  days  or 
weeks  before. 

Thus  it  is  that  three  means  of  conducting  the  recitation  in 
Arithmetic  should  bo  iised  by  the  teacher  :   first,  examination  to 


teachers'  manual.  5 

test  the  pupils'  knowledge  of  the  topic  or  topics ;  secondly,  teach- 
ing to  awaken  new  ideas  in  the  pupils'  minds ;  and  thirdly, 
drilling  to  fix  and  strengthen  the  pupils'  knowledge  of  what  has 
been  taught.  These  processes  are  not  always  separated  in  the 
recitation.  Sometimes  in  the  teaching  exercise  there  is  more  or 
less  of  examination  and  drill,  and  sometimes  in  the  examination 
there  is  necessarily  much  useful  drill.  But  they  all  have  their 
j)la(H'  in  the  recitation,  and  no  one  of  them  should  be  neglected. 

Methods  of  Drill.  —  The  effectiveness  of  the  drill  exercise  will 
depend  largely  upon  the  methods  employed.  The  prime  object  of 
the  drill  is,  as  has  been  said,  to  fix  in  the  mind  what  has  been 
taught.  To  gain  this  object,  there  should  be  a  repetition  of  the 
mental  act  which  was  occasioned  when  the  principle  or  process 
was  taught.  In  the  drill  exercise,  the  teacher  should  have  in  mind 
exactly  what  acts  of  the  mind  are  to  be  repeated,  and  he  should 
also  aim  to  have  every  pupil  employed  during  the  exercise.  Drill 
which  has  not  a  definite  object,  or  which  consists  of  a  rejietition  of 
words  only,  or  which  permits  inactivity  on  the  part  of  some 
members  of  the  class,  cannot  accomplish  all  the  desired  ends. 

The  drill  may  be  botli  oral  and  written.  If  oral,  the  given 
problem  may  be  read  by  one  pupil,  all  the  rest  of  the  class  reading 
it  silently  at  the  same  time ;  or  it  may  be  dictated  by  the  teacher. 
In  either  case  all  should  solve  the  problem  together,  either  follow- 
ing the  oral  solution  of  a  l)U})il,  or  else  performing  it  silently  and 
indicating  the  answer  on  slate  or  paper.  By  the  latter  plan  it 
would  be  well  to  have  all  tlie  pupils  work  and  write  very  promptly 
exactly  at  the  same  time.  For  example,  the  pupils  have  before 
them  paper  or  slate  and  pencil.  The  teacher  says,  "pencil  in 
hand,"  and  dictates  (not  from  a  book)  the  problem.  After  giving 
sufficient  time  for  all  to  solve  it,  the  teacher  says,  "Avrite  the 
answer,"  and  then  after  just  enough  time  for  the  pupils  to  write 
the  figures  of  the  answer,  says,  "  pencils  down."  The  teacher  then 
gives  the  answer,  and  asks  all  pujjils  wlio  have  that  answer  to  hold 
up  the  slate  or  paper  so  that  tlie  teacher  can  see  the  answer.  It 
will  be  necessary  for  the  figures  of  the  answer  to  be  written  large 


6  GRADED   ARITHMETIC. 

and  in  a  given  place,  that  they  may  be  seen  readily.  Properly 
managed,  this  method  of  drill  Avill  afford  opportunity  for  giving 
much  practice  in  a  short  period  of  time.  This  method  will  serve 
for  examination  as  well  as  for  dictation. 

Written  drill  work  may  be  given  from  dictation  or  from  the 
book.  If  from  the  book,  a  number  of  exercises  may  be  given 
out  for  pupils  to  perform  as  many  as  they  can  within  a  certain 
time,  or  the  exercises  may  be  given  singly  to  members  of  the  class, 
each  pupil  performing  such  problems  as  are  most  needful  or  most 
useful  for  him  to  perform. 

Reviews.  —  For  the  sake  of  making  pupils  familiar  with  certain 
important  principles  or  processes,  and  also  for  the  sake  of  connect- 
ing together  kindred  subjects,  fre(j[uent  reviews  should  be  made. 
In  giving  such  reviews,  let  the  teacher  have  in  mind  a  well-defined 
purpose,  and  not  force  pupils  to  work  simply  for  the  sake  of  work- 
ing. No  exercise  should  be  given  that  is  not  needed,  either  for 
fixing  in  the  mind  what  has  been  taught,  or  for  testing  the  pupils' 
knowledge  of  a  given  subject,  or  for  securing  skill  and  facility  in 
the  use  of  numbers. 

Tasks  and  Desk-work.  —  The  lessons  set  for  practice  or  study 
should,  as  nearly  as  possible,  serve  the  needs  of  every  pupil  in  the 
class.  So  far  as  the  character  of  the  work  is  concerned,  those  sub- 
jects only  should  be  given  for  tasks  which  are  really  needed  to  be 
reviewed  or  which  will  serve  as  a  test  of  the  pupils'  knowledge. 
The  given  lesson  also  should  be  of  sufficient  length  to  employ  all 
the  pupils  of  the  class  a  reasonable  amount  of  time.  To  accom- 
modate the  work  to  the  different  degrees  of  ability  and  quickness 
in  members  of  the  class,  it  may  be  well  to  give  a  lesson  which  all 
can  do,  and  in  addition,  extra  Avork  for  those  Avho  are  able  to  do 
more. 

Illustrations.  —  Problems  involving  measurements,  and  other 
problems  whose  conditions  are  not  quite  cleav.  should  l)r  illustrated 
by  means  of  ])lans  or  diagrams.  Care  should  be  taken,  however, 
that  the  attention  of  pupils  !)(>  not  diverted  fi'om  the  real  purpose 
of  the  drawings  by  their  nicety  or  elaborateness,  and  that  illustra- 


TEACHEKS     MANUAL.  7 

tions  be  not  required  when  the  problems  are  fully  understood,  or 
when  the  problems  can  be  performed  without  them. 

Reading  Problems.  —  Let  the  pupils  form  the  habit  from  the 
outset  of  reading  over  thoughtfully  every  problem  before  they 
begin  the  solution.  The  points  to  be  noted  in  the  reading  are  : 
What  is  required  ?  What  steps  are  to  be  taken  ?  About  how  large 
will  the  answer  ho  ? 

Analyses  and  Explanations.  —  Oral  analyses  or  explanations  in 
the  lower  grades  should  be  very  simple.  In  these  grades  short 
and  direct  statements  as  to  what  has  been  or  is  to  be  done  in  the 
solution  of  a  given  problem  may  be  made.  The  statements  should 
as  nearly  as  possible  represent  exactly  the  pupil's  thought  and 
should  consist  therefore  of  words  largely  of  liis  own  choice  and 
arrangement.  In  the  middle  and  upper  grades  more  and  more 
minute  analyses  may  be  insisted  upon,  the  pupils  being  required  as 
their  minds  develop  to  give  reasons  for  the  steps  taken.  But  in 
no  part  of  the  course  should  pupils  be  required  to  give  formal 
and  ready-made  explanations  which  do  not  represent  their  own 
thinking.  Some  statements  which  explain  to  the  teacher  may 
actually  bewilder  the  pupils  and  deaden  their  thinking  powers. 

Problems  and  Indicated  Operations.  —  From  a  very  early  period 
in  the  course  in  Arithmetic  the  pupils  should  be  led  to  see  and  to 
indicate  operations  that  are  involved  in  problems.  In  the  first 
year  pupils  are  encouraged  to  make  story  problems  from  equations 
and  soon  after  to  make  the  equations  from  the  story  problems. 
This  work  should  go  on  all  through  the  course  until  in  the  later 
grades  much  of  the  work  in  Arithmetic  may  well  consist  of  written 
or  oral  statements  of  processes  that  are  involved  in  given  problems. 

Forecasting  Answers. — ^  Every  teacher  knows  what  surprising 
and  unreasonable  ansAvers  to  problems  are  given  by  pupils,  being 
in  some  cases  many  thousand  times  too  large  or  small.  To  prevent 
such  mistakes  and  to  encourage  thoughtful  attention  to  the  condi- 
tions of  problems,  it  is  well  by  means  of  questions  first  to  give  an 
approximate  estimate  as  to  what  the  answer  will  be.  Suppose 
for  example  the  problem  is  :  "  How  many  bushels  of  potatoes  will 


8  GEADED   ARITHMETIC. 

be  gathered  from  a  lot  of  land  100  feet  square  if  it  yields  at  the 
rate  of  200  bushels  to  the  acre  ?  "  The  teacher  may  ask  :  "  How 
many  sq.  ft.  in  the  lot  ?  What  part  of  an  acre  in  the  lot  ?  If  the 
lot  is  a  little  less  than  one  quarter  of  an  acre,  less  than  how  many 
bushels  will  the  lot  yield?"  Or  if  the  problem  is  :  "How  many 
bushels  of  wheat  can  be  put  into  a  bin  6  ft.  4  in.  long,  4  ft.  2  in. 
wide  and  3  ft.  deep,"  the  teacher  may  ask  :  *'  About  how  many 
cu.  ft.  does  the  bin  contain  ?  How  do  a  bushel  and  a  cubic  foot 
compare  ?  Will  the  bin  contain  a  few  more  or  less  than  72 
busliels  then  ? "  Gradually  the  teacher  may  give  less  and  less 
assistance  in  this  direction  until  the  pupils  will  be  able  to  forecast 
the  answers  independently  and  form  the  habit  of  so  doing. 

Languag^e.  —  The  good  teacher  is  quick  to  see  the  useful  and 
constant  opportunities  which  Arithmetic  affords  for  teaching  lan- 
guage. From  the  very  first  year  when  children  are  encouraged  to 
tell  little  number  stories  to  the  highest  grade  of  the  Grammar 
School  when  explanations,  rules  and  definitions  are  required  to  be 
made  by  the  pupils  themselves,  is  there  constant  occasion  for  the 
clear  and  accurate  expression  of  thought.  The  very  accuracy  of 
the  thought  required  in  Arithmetical  processes  and  definitions 
makes  accuracy  of  expression  necessary  both  in  the  choice  and  in 
the  arrangement  of  words. 


SECTION  II. 

FIRST   STEPS   IN   NUMBER. 

Before  the  book  is  put  into  the  hands  of  children,  it  will  be  well 
to  teach  the  numbers  to  ten  by  the  aid  of  objects  without  the  use 
of  figures.  This  may  occupy  a  greater  or  less  time  according  to 
the  ability  and  previous  training  of  the  children.  Most  children 
who  enter  school  at  five  years  of  age  know  very  little  of  number, 
while  others  who  enter  at  .a  later  period,  or  who  have  had  the 


TEACHEES'    MANUAL.  9 

advantage  of  a  good  inheritance  or  helpful  home  and  kindergarten 
training,  can  recognize  numbers  and  make  all  combinations  and 
separation  of  numbers  to  four  or  six.  For  the  sake  of  a  proper 
adaptation  of  the  work  to  the  capacity  of  the  children,  the  separa- 
tion of  the  children  into  groups  will  be  found  advisable.  The 
division  made  need  not  be  fixed.  There  is  as  much  difference 
among  children  in  the  growth  of  their  intelligence  as  there  is  in 
the  intelligence  itself.  Accordingly  pupils  should  be  transferred 
from  one  group  to  another  as  soon  as  they  are  discovered  to  be 
ahead  of  or  behind  their  fellows. 

Counters.  — •  On  some  accounts  the  variety  of  objects  used  for 
teaching  numbers  in  the  first  lessons  should  not  be  great,  being 
limited  to  the  splints  or  wooden  blocks.  If  blocks  are  used,  they 
might  be  either  of  three  sizes,  viz.:  cubes  (1X1X1  in.),  half-cubes 
(1  X  1  X  |-  in.),  quarter-cubes  (1  X  ^  X  -g-  in.).  The  whole  cubes 
are  preferred  by  many  teachers.  Splints  have  the  advantage  of 
being  easily  handled. 

Methods.  —  For  teaching  number  to  young  children,  there  should 
be  a  table  about  which  the  children  can  stand  in  full  sight  of  the 
counters  in  front  of  the  teacher,  who  sits  at  the  head  of  the  table. 
Or,  if  there  is  no  table,  the  children  may  sit  in  their  seats  and  the 
teacher  use  an  inclined  table  in  sight  of  the  children. 

If  the  number  three  and  their  combinations  are  known  by  the 
children,  ask  them  to  place  three  blocks  together.  After  this  is 
done  by  all,  the  teacher  and  children  put  one  block  with  the  three 
blocks,  and  the  new  number  is  named  by  the  teacher.  Teacher: 
Three  blocks  and  one  block  are  how  many  blocks  ?  Children 
(answering  singly):  Three  blocks  and  one  block  are  four  blocks. 
T.  (taking  away  one  block):  Four  blocks  less  one  block  are  how 
many  blocks  ?  Ch.  (taking  away  one  block  as  the  teacher  did) : 
Four  blocks  less  one  block  are  three  blocks.  T.  (moving  the 
blocks) :  Two  blocks  and  two  blocks  are  how  many  blocks  ? 
Four  blocks  less  two  blocks  ?  One  block  and  three  blocks  ? 
Four  blocks  less  three  blocks  ?  Two  twos  of  blocks  ?  One  four 
of  blocks  ?     One-half  of  four  blocks  ?     One-fourtli  of  four  blocks  ? 


10 


GRADED   ARITHJMETIC. 


How  many  twos  of  blocks  in  four  blocks  ?  How  many  ones  of 
blocks  in  four  blocks  ?  Answers  to  these  questions  are  to  be  given 
in  order,  the  objects  being  used  in  the  manner  indicated.  The 
work  with  objects  should  be  continued  until  the  children  have  a 
clear  and  definite  idea  of  the  combinations.  These  exercises  may 
be  followed  by  a  comparison  of  four  blocks  with  three  blocks,  and 
two  blocks  and  one  block,  the  children  being  led  with  the  groups 
before  them  to  answer  the  following  questions  :  Four  blocks  are 
how  many  more  than  three  blocks  ?  Three  blocks  are  how  many 
less  than  four  blocks  ?  Four  blocks  are  how  many  more  than  two 
blocks  ?  Two  blocks  are  how  many  less  than  four  blocks  ?  Four 
blocks  are  how  many  more  than  one  block  ?  One  block  is  how 
many  less  than  four  blocks  ? 

In  the  same  way  numbers  to  ten  may  be  taught,  no  abstract 
memory  work  being  required  or  expected  during  these  first  lessons. 
The  following  facts  of  numbers  to  ten  should  be  taught : 


^      r4  +  l;    5-1;    3  +  2;    5-2;    2  +  3;    5' 
\  5  —  1 ;    1  five  ;    5  ones  ;  5  -i- 1 ;    ^  of  5. 


3;    1  +  4; 


r^  +  i 

6  —  4 
6  —  2 


6  —  1;    4  +  2;    6  —  2;    3  +  3;    6  —  3;    2  +  4; 

1  +  5  ;    6  —  5  ;    1  six  ;    6  ones  ;    6  -i-  1  ;    3  twos  ; 

2  threes  ;    6^3;    i  of  6  ;    ^  of  6  ;    ^  of  6. 


:  -< 


6  +  1;    7-1;    5  +  2;    7-2;    4  +  3;    7-3;    3  +  4; 

7  —  4;    2  +  5;    7  —  5;    1  +  6;    7  —  6;    1  seven  ;    7  ones  ; 
7^1;    }  of  7. 


8:^ 


7  +  1;    8-1;    6  +  2;    8-2;    5  +  3;    8-3;    4  +  4; 
8-4;    3+5;    8-5;    2  +  6;    8-6;    1  +  7;    8-7; 
1  eight ;    8  ones  ;    8  -i-  1  ;    4  twos  ;    8  -j-  2  ;    2  fours  ; 
L8-^4;    ^of  8;    1  of  8;    ^  of  8, 


TEACHERS     MANUAL. 


11 


10 


f8  +  1;    9-1;  7  +  2;    9-L>;    C.  +  3  ;    9-3;    5  +  4; 

9-4;    4  +  5;  9-5;    11  +  G  ;    9-G;    2  +  7;    9-7; 

1  +  8  ;    9  —  8  ;  1  nine  ;   9  ones  ;   9^1;   3  threes  ;  9  -i-  3  ; 
.  1-  of  9  ;    ^  of  9. 

9  +  1  ;    10  -  1  ;    8  +  2  ;    10  -  2  ;    7  +  3  ;    10  -  3  ;    T)  +  4  ; 

10  —  4  ;    5  +  5  ;    10  —  5  ;    3  +  7  ;    10  —  7  ;   2  +  8  ;   10  —  8  ; 
1  +  9  ;    10  —  9  ;    1  ten  ;    10  ones  ;    10  -4-  1  ;    5  twos  ; 
10^2;    2  fives  ;    10  -^  5  ;    ^  of  10  ;    l  of  10  ;    -jL  of  ]  0. 


Picturing  of  Problems.  —  The  picturing  of  problems  may  some- 
times be  given  for  busy-work  in  the  lower  grades,  one  object  being 
to  provide  a  pleasant  inducement  for  the  children  to  review  what 
has  been  taught.  This  suggests  the  advisability  of  teachers'  show- 
ing pupils  hov/  the  drawings  may  be  made  and  allowing  them  a 
certain  degree  of  freedom  for  the  exercise  of  their  inventive  powers. 
But  care  should  be  taken  not  to  carry  the  picturing  too  far.  It 
should  be  remembered  that  such  work  is  a  means  and  not  an  end. 
Properly  used,  the  representations  will  serve  to  make  clear  the 
relations  which  are  not  fully  grasped  and  to  fix  in  the  mind  com- 
binations and  processes  which  are  not  quite  clear  ;  but  carried  to 
the  extent  of  diverting  the  attention  from  the  main  purpose  of  the 
exercise,  they  are  to  be  deprecated. 

TJse  of  Figures.  —  Early  in  the  year  figures  to  represent  numbers 
should  be  used,  first  in  having  the  children  read  Avhat  is  written 
by  the  teacher,  and  afterwards  in  writing  the  figures.  Such  work 
may  proceed  in  the  following  order  :  (1)  The  objects  counted  and 
placed  in  a  group  by  each  child.  (2)  The  name  of  the  number 
given  by  the  teacher  and  repeated  by  the  children.  (3)  The  figure 
written  on  the  board  and  read  by  the  children.  (4)  The  figure 
copied  or  written  from  dictation  on  slate  or  paper  by  each  child. 
The  teacher  should  be  careful  to  present  only  good  models  before 
the  children  for  imitatioij. 

The  writing  of  equations  to  represent   combinations  of  objects 


12  GRADED   ARITHMETIC. 

and  the  interpretation  of  represented  combinations  constitute  a 
useful  feature  of  number  work.  For  example,  the  children  are  told 
to  put  3  sticks  with  3  sticks,  and  after  telling  how  many  in  all,  the 
teacher  writes  on  the  board  3  -[-  3  =  6,  which  the  children  read. 
The  same  is  done  after  all  operations  with  sticks  in  addition,  sub- 
traction, multiplication  and  division.  At  another  time  the  teacher 
writes  on  the  board  an  expression,  as  4  +  2,  and  asks  the  children 
to  do  with  the  sticks  what  that  says.  This  exercise  will  lead  to 
a  good  kind  of  busy  work  in  number. 

Number  Stories.  —  For  purposes  of  language  as  well  as  for  the 
purpose  of  affording  a  pleasant  means  of  drill  in  number,  pupils 
should  be  led  to  tell  stories  involving  the  combinations  which  have 
been  taught.  Tlie  general  plan  of  such  work  may  be  as  follows  : 
First  let  the  teacher  make  a  story,  each  pupil  using  the  sticks  as 
called  for.  Then  each  pupil  may  tell  a  story  and  as  he  talks,  he, 
with  the  other  members  of  the  class,  uses  the  objects  for  illustra- 
tion. After  this  has  been  done  with  the  aid  of  sticks,  stories  may 
be  told  from  spoken  or  written  examples  or  equations,  and  finally 
they  may  be  told  with  no  assistance  or  suggestion  from  any  one. 
In  all  this  work  the  aim  should  be  to  make  it  so  interesting  that 
all  members  of  the  class  will  pay  close  attention.  The  following 
lesson  will  illustrate  the  plan  above  given  : 

Teacher.  Show  me  six  sticks.  Sliow  me  four  of  them.  How 
many  sticks  are  four  sticks  and  two  sticks  ?  Call  the  sticks  books. 
ChildreM.  Four  books  and  two  books  are  six  books.  T.  I  will  tell  a 
story  and  as  I  tell  the  story  you  may  work  with  the  sticks.  I  have 
four  apples  and  my  brother  has  two  apples.  How  many  apples  have 
we  both  together  ?  C.  Both  have  four  apples  and  two  apples.  Four 
apples  and  two  apples  are  six  apples.  T.  Who  can  make  a  story 
about  five  and  one  ?  Mary.  M.  I  had  five  cents  and  my  mother 
gave  me  one  cent.  How  many  cents  have  I  now  ?  T.  Charlie  may 
tell.  C.  You  have  five  cents  and  one  cent.  Five  cents  and  one 
cent  are  six  cents. 

(The  "  story  "  may  be  given  in  the  form  of  a  question,  as  above, 
or  the   children  may  be   led  to  give  the   solution  in  a  statement, 


teachers'  manual.  13 

thus  :  If  I  have  5  cents  and  my  mother  gives  me  1  cent  more,  I 
have  5  cents  and  1  cent.     5  cents  and  1  cent  are  6  cents.) 

T.  Who  Avill  tell  a  story  about  3  twos  ?  Jamie.  J.  If  I  buy  3 
pencils  and  pay  2  cents  apiece  for  them  I  pay  3  times  2  cents. 
3  times  2  cents  are  6  cents.  T.  Willie,  another.  W.  3  rabbits 
have  3  times  2  eyes.     3  times  2  eyes  are  6  eyes. 

In  all  this  work  it  is  supjjosed  that  every  member  of  the  class  is 
following  with  or  without  the  objects  the  stories  as  they  are  told. 
More  elaborate  stories  may  be  told  and  more  extended  combinations 
may  be  made  if  the  attention  can  be  held.  Such  questions  as  the 
following  may  not  be  too  difficult  provided  the  objects  are  used 
and  progress  is  made  by  easy  steps. 

Charlie  was  given  10  cents  to  buy  something  for  his  mother. 
He  bought  2  oranges  at  3  cents  apiece  and  a  yeast-cake  for  2  cents. 
How  many  cents  did  he  bring  back  ? 

I  had  6  cents  and  my  brother  gave  me  2  more.  With  the  money 
I  had  then  I  bought  apples  at  two  cents  apiece.  How  many 
apples  did  I  buy  ? 

From  given  stories  children  may  be  led  to  perform  the  required 
operations  with  objects  and  to  make  the  proper  equations.  The 
finding  and  writing  of  equations  from  problems  constitute  an  im- 
portant part  of  luimber  work  later  in  the  course  and  may  well  be 
begun  in  the  lowest  grade  of  schools. 


SECTION   III. 

NOTES   FOR   BOOK   NUMBER   ONE. 

(The  numbers  at  the  head  of  the  page  indicate  tlie  number  and  page  of  the 
Pupils'  Book  for  which  notes  are  made  :  thus  [II.  18]  means  Book  No.  2, 
18th  page.  Figures  in  heavy  type  on  a  page  of  the  Manual  denote  the  page  or 
pages  of  the  book  for  which  notes  are  made.) 

The  exercises  of  Section  I.  are  intended  as  a  review  of  what  has 
been  taught  with  objects.  It  may  be  necessary  still  for  objects  to 
be  used  occasionally,  both  by  pupils  at  their  seats  and  by  the 


14  G:fcADEt)   ARITHMETIC.  [1.  2 

teacher  and  pupils  in  recitation,  but  the  aim  should  be  to  lead 
pupils  to  perform  promptly  the  exercises  of  this  section  without 
objects  before  the  second  section  is  begun.  If  an  example  or 
problem  is  not  readily  performed,  the  want  of  promptness  is  due 
to  one  of  two  causes  :  either  the  pupils  do  not  understand  the 
operation  involved,  or  else,  understanding  it,  they  have  not  facility 
in  performing  it.  If  an  operation' is  not  understood,  it  should  be 
taught,  and  the  best  method  of  teaching  ideas  of  numbers  and  their 
operations  is  by  the  use  of  objects.  Facility  is  gained  through 
continued  practice  in  repeating  the  operation  in  which  facility  is 
needed.  This  practice  may  be  gained  in  the  recitation  and  in  the 
busy  work  of  the  pupils  at  their  seats.  If  the  previous  lessons 
have  been  well  taught,  it  is  siipposed  that  pupils  need  only  the 
drill  of  busy  work  and  recitation  practice  in  exercises  of  Section  I. 
to  prepare  them  for  the  work  of  Section  II. 

For  counters  use  the  blocks  or  wooden  sticks  which  have  been 
used  previously  (see  Manual,  page  9),  or  small  pasteboard  squares 
which  can  be  made  with  little  difficulty.  If  the  recitation  table 
could  have  upon  it  lines  indicating  squares  of  the  size  of  the 
counters,  some  advantages  for  placing  and  comparing  the  counters 
would  be  gained. 

The  use  of  lines  and  the  rough  drawings  of  objects  in  perform- 
ing problems  is  shown  on  pages  20  and  26.  Such  work  should 
be  used  constantly  in  the  lower  grades,  care  being  taken  that  in 
representing  numbers  and  operations  by  drawing  the  attention  of 
children  be  not  diverted  from  the  real  purpose  of  the  exercise, 
either  in  the  complexity  of  the  representations  or  in  their  extreme 
accuracy  or  nicety. 

Other  suggestions  as  to  the  scope  and  use  of  the  book  will  be 
found  in  the  "Note  to  Teachers,"  page  1. 

2—11  The  design  of  these  exercises  is  to  present  graphically 
before  the  children  the  various  combinations  and  to  give  a  pleasant 
means  of  reviewing  what  they  have  had  with  objects.  Let  the 
children  practice  upon  the  illustrated  pages  until  the  combinations 
can  be  given  quite  freely.     The  sign  of  multiplication  as  used  in 


I.  12]  teachers'  manual.  15 

these  books  is  supposed  to  be  read  "multi})lied  by "  rather  than 
"  times  "  for  two  reasons  :  1st,  for  the  sake  of  uniformity.     The 

signs  -| and  -7-  all  indicate  that  the  number  preceding  the  sign 

is  to  be  operated  upon,  2  +  3  means  that  3  is  to  be  added  to  2  and 
that  the  expression  should  be  read  2  and  3  or  2  plus  3 ;  6  —  4 
means  that  4  is  to  be  taken  out  of  6  and  it  should  be  read  G  less  4  ; 
8-r-4  means  that  8  is  to  be  divided  by  4  and  should  be  so  read. 
In  the  same  way  4X2  means  that  4  is  to  be  multiplied  by  2 
and  that  it  be  read  4  multiplied  by  2.  Another  and  stronger  reason 
for  placing  the  multiplier  after  the  sign  is  convenience  in  the 
analysis  of  problems.  This  will  appear  later  when  the  written 
analyses  of  problems  are  made  by  the  pupils.  If  desired  the 
expression  4X2  may  be  read  2  times  4. 

The  pages  opposite  the  illustrative  work  are  designed  for  busy- 
work  and  for  recitation.  In  recitation  the  children  should  be  led 
to  read  the  sentences  supplying  the  ellipses  as  they  read. 

13-15  These  exercises  are  to  be  used  for  busy-work  and  also 
for  recitation.  If  necessary  the  children  may  be  permitted  to  use 
the  splints  in  finding  the  answers,  but  in  the  recitation  they  should 
be  expected  to  give  the  answers  promptly  after  the  statement,  thus 
"4t  and  5  are  9,"   "3  and  5  are  8." 

16  A  good  exercise  is  to  put  pairs  of  numbers  on  the  board  as 
here  indicated  and  have  the  children  give  the  sum  instantly.  This 
will  assist  in  subsequent  work  in  adding  columns. 

1 7  The  children  should  be  encouraged  to  make  illustrations  of 
processes  as  here  indicated. 

18—19  The  children  with  a  little  assistance  can  be  led  to  make 
original  stories  like  the  following  :  "  My  father  had  8  ducks  and 
he  sold  4  of  them.     He  had  then  4  ducks  left." 

20  Good  illustrative  work  is  here  shown  which  the  children 
can  occasionally  do  bv  themselves  for  busy-Avork. 

31—35  A  variety  is  here  given  for  drill  both  in  busy-work  and 
recitation. 

36—38  With  little  effort  on  the  part  of  the  teacher,  the 
children  can  be  led  to  illustrate  problems  for  themselves.     But  in 


16  GRADED   ARITHMETIC.  [I.  29 

all  this  work  observe  the  caution  given  on  pages  11  and  14  of  the 
Manual. 

39—36  Let  the  teaching  of  each  number  above  10  precede  the 
recitation  from  tlie  book.  Thus  in  teaching  11,  place  upon  the 
table  in  sight  of  all  the  pupils  10  square  blocks  in  a  column.  The 
children  who  are  sitting  at  their  seats  or  standing  around  the  table 
do  the  same  with  their  squares.  The  teacher  asks  how  many 
blocks  there  are  and  proceeds  as  follows  :  "We  will  place  one 
more  block  with  the  ten.  Now  we  have  eleven  blocks.  How  many 
blocks  have  you,  Charley  ?  Eddie  ?  etc.  How  many  blocks  are  ten 
blocks  and  one  block,  Mary  ?  Susie  ?  etc.  Take  away  one  block 
and  how  many  have  you  ?  Eleven  blocks  less  one  block  are  how 
many  blocks  ?  "  Other  facts  in  11  developed  in  the  same  way  are 
shown  on  page  29.  Where  there  are  two  subtractions,  as  in 
11  —  5  —  5,  lead  the  children  first  to  combine  5  -\-  o  -\-l,  leaving 
the  blocks  as  indicated,  and  then  to  remove  a  little  way  from  the 
collection  5  blocks,  covering  them  up  as  they  say  :  "  11  blocks  less 
5  blocks  are  6  blocks  " ;  repeating  the  process  they  say  :  "  6  blocks 
less  5  blocks  are  1  block,  11  blocks  less  5  blocks  less  5  blocks  are 
1  block."  The  groups  of  blocks  are  now  in  position  to  divide  by  5, 
the  question  being  :  "  In  11  there  are  how  many  fives  and  how 
many  left  over  ?"  or,  '-5  blocks  in  11  blocks  how  many  times  and 
how  many  left  over  ? "  After  the  separation  of  the  number  into 
fours,  threes,  and  twos,  a  comparison  of  11  with  all  numbers  below 
should  be  made  ;  thus,  in  answer  to  questions,  the  children  are  led 
to  say:  ''11  blocks  are  1  more  than  10  blocks  ;  11  blocks  are  2 
more  than  9  blocks,  etc.     11  is  1  more  than  10,"  etc. 

The  writing  and  reading  of  numbers  should  be  carried  on  in 
connection  with  objects.  Sticks  may  be  the  most  convenient 
objects  for  this  purpose.  The  teacher  takes  11  sticks  and  after 
asking  the  children  how  many  there  are,  binds  10  of  them  together 
with  a  rubber  band  and  says  :  This  is  1  ten  of  sticks.  How  many 
tens  of  sticks  in  eleven  and  how  many  more  ?  The  cliildren  should 
be  led  to  bind  the  10  sticks  into  a  ten  as  the  teacher  has  done  and 
to  say  that  in  11  there  is  1  ten  of  sticks  and  1  more.     To  express 


1. 37]  teachers'  manual.  17 

in  figures  this  number  the  ten  bundle  may  be  placed  on  the  ledge 
of  the  blackboard  and  the  children  be  led  to  write  1  above  it  for 
1  ten.  The  1  stick  may  be  placed  beside  the  ten  and  another 
figure  1  be  placed  above  the  1  stick,  making  1  ten  and  1,  or  11. 
This  exercise  should  be  worked  out  very  slowly  and  carefully,  and 
if  necessary  it  should  be  repeated  several  times,  the  purpose  being 
to  lead  the  children  to  liave  a  clear  idea  of  the  decimal  system  at 
the  outset. 

After  this  work  with  objects  has  been  done,  equation  statements 
may  be  given  by  the  pupils  in  answer  to  questions  given  by  the 
teacher  or  read  from  the  book.  Thus  all  operations  that  have  been 
performed  with  the  objects  are  uoav  given  orally  without  objects. 
Then  follows  the  writing  of  the  equations  by  the  children.  This 
will  constitute  a  part  of  the  busy-work  of  the  children,  who  copy 
from  the  book  the  questions  and  complete  the  equations.  The 
story  problems  which  naturally  follow  may  be  told  and  solved  in 
the  recitation  and  also  at  the  seat. 

The  order  of  procedure  in  all  tins  work  should  be  noticed — 
(1)  operations  with  objects,  (2)  giving  of  oral  equations,  (3)  giving 
of  written  equations,  (4)  giving  of  story  problems.  Sometimes 
drawings  may  be  made  in  illustration  of  operations  directly  after 
the  objects  have  been  used,  and  sometimes  the  drawings  may  be 
used  instead  of  the  objects.  For  review,  the  order  above  given  may 
be  changed  or  inverted.  The  oral  or  written  equations  may  be 
given  by  the  children,  who  will  illustrate  or  demonstrate  the  equa- 
tions by  the  use  of  objects  or  drawings  ;  or  the  children  may  give 
the  story  problem,  afterwards  solve  it  by  objects  or  drawings  and 
finally  make  the  oral  and  written  equations. 

The  order  above  given  with  such  variation  as  circumstances  will 
determine,  should  be  followed  in  treating  eacli  new  number. 

37-39  Nearly  all  the  questions  on  these  pages  should  be  solved 
by  means  of  simple  drawings  as  shown  on  page  87.  Some  care 
will  have  to  be  taken  to  show  the  children  just  what  is  expected  of 
them.  Slow  progress  in  picturing  problems  at  this  stage  must  be 
expected.     From  what  the  teacher  does   upon  the  blackboard  in 


18  GRADED   ARITHMETIC.  [I.  40 

illustrating  a  process,  and  from  little  corrections  that  are  made, 
the  children  will  gradually  get  the  idea  of  representing  the  condi- 
tions and  the  solution  of  problems  by  means  of  drawings. 

40  In  teaching  13,  iirst  use  the  squares  and  blocks  in  the  order 
indicated,  somewhat  as  is  given  on  pages  29  and  33.  With  some 
suggestions  and  directions,  the  children  can  be  led  to  draw  the 
squares  and  to  indicate  what  process  is  shown.  Dwell  upon  this 
page  until  a  tolerable  degree  of  familiarity  with  the  combinations 
is  had. 

4:1—43  In  reciting  these  exercises  the  children  may  read  only 
the  answer  thus  :  in  a,  page  42,  "  Twelve  boys  and  one  boy  are 
thirteen  boys,"  etc.;  in  e,  "In  twelve  there  are  6  twos,"  etc.;  and 
in  g,  "  I  must  put  ten  with  three  to  make  thirteen."  In  the  story 
problems  let  the  children  first  read  the  problem  and  then  give  the 
answer  in  a  good  statement,  tlius  :  in  /,  page  43,  "  The  little  boy's 
sister  found  eight  eggs  and  three  eggs,  or  eleven  eggs." 

44  A  teaching  exercise  with  the  coins  should  precede  the 
recitation  of  these  problems.  10  one-cent  pieces,  5  two-cent  pieces, 
2  five-cent  pieces,  and  a  dime  will  be  needed  to  show  all  the  facts 
given.  The  children  already  have  some  knowledge  of  the  value 
of  these  coins,  and  the  exercise  with  them  may  be  made  very 
interesting.  Thus,  the  teacher  may  say,  "  Who  will  find  six  cents 
from  these  coins  ?  "  "  Charlie  has  a  dime.  Who  will  take  from 
the  two-cent  pieces  enough  to  have  as  much  money  as  he  has  ?  " 
The  children  will  also  enjoy  buying,  selling,  and  making  change 
with  the  coins. 

45  The  combinations  indicated  on  this  page  should  be  taught 
first  with  objects,  and  then  shown  by  squares  as  given  on  pages  29 
and  33.  The  children  should  also  be  expected  to  make  the  draw- 
ings with  as  little  assistance  as  possible. 

46—47  In  the  column  addition,  lead  the  children  to  add  in 
pairs,  thus:  in  Ex.  4  :  6,  8  ;  7,  11,  etc.;  and  in  Ex.  5:  6,  13  ; 
8,  12,  etc. 

49  First  teach  with  objects  as  before.  The  answers  should  be 
given  with  and  without  the  aid  of  squares. 


1. 51]  teachers'  manual.  19 

51-53  The  form  of  problems  and  answers  in  Exercises  12  and 
13  is  a  model  for  original  problems  which  should  be  given  by  the 
cliildren  occasionally.  These  may  be  given  with  and  without 
objects. 

54  The  problems  involving  days  in  the  week,  weeks  in  the 
month,  and  units  in  the  dozen,  should  be  extended  by  the  teacher 
and  by  the  children  with  as  great  variety  as  possible. 

iiiy  Introduce  the  measures  here,  using  water  or  dry  sand.  Let 
the  children  measure  the  water  or  sand  until  the  facts  become  fully 
understood.  The  measures  should  also  be  used  in  teaching  prob- 
lems similar  to  those  in  the  lower  part  of  this  page.  This  is 
followed  by  picturing  the  problems  as  shown  in  the  cut.  The 
children  should  make  the  drawings  Avitli  as  little  assistance  from 
the  teacher  as  possible.  Such  work  will  be  found  very  useful  and 
interesting  to  the  children.  The  illustrations  should  always  be 
made  whenever  the  children  find  difficulty  in  making  a  mental 
image  of  the  conditions,  and  in  performing  the  problem.  Three  or 
four  days  may  be  profitably  spent  upon  problems  similar  to  those 
given  on  this  page  and  the  page  following. 

57  The  measuring  strip  called  for  should  be  cut  from  stiff 
paper ;  pasteboard  or  cardboard  would  be  better.  If  the  children 
cannot  cut  and  mark  this  strip  accurately,  the  teacher  or  older 
children  should  do  it.  Several  exercises  with  this  measure  and 
a  yard-stick,  similar  to  those  on  the  lower  part  of  the  page,  should 
be  given. 

58  A  teaching  exercise  should  precede  drill  upon  this  page. 
Teach  rectangle  and  square,  by  showing  that  rectangles  have 
square  corners  and  that  squares  are  rectangles  whose  sides  are 
equal.  Paper  cutting  and  folding  should  accompany  the  solution 
of  c,  d,  e,  f,  and  g.     Other  similar  problems  may  be  given. 

59  If  the  children  cannot  perform  these  problems  readily, 
objects  and  drawings  should  be  used.  Let  the  answers  be  in 
entire  sentences,  thus  :  "  In  two  quarts  there  are  four  pints,"  or 
"There  are  four  pints  in  two  quarts."  Do  not  insist  upon  one 
form   of  answer   to   these   problems.      If  the    statement   of  the 


20  GRADED   ARITHMETIC.  [I.  62 

pupil  expresses  a  clear  thought  of  the  number  combinations  let 
it  pass. 

62  The  children  should  be  encouraged  to  illustrate  these  prob- 
lems in  as  many  ways  as  possible.  After  the  processes  have  been 
shown  by  drawings,  the  problems  should  be  read  and  full  state- 
ments of  answers  should  be  given.  Here  also  one  particular  form 
of  statement  should  not  be  insisted  upon. 

64  Some  of  these  problems,  such  as  d,  f,  g,  and  li,  should  be 
taught  before  they  are  given  to  the  children.  Add  by  pairs  as  well 
as  singly,  numbers  indicated  in  the  columns.  Thus  the  first 
column  may  be  added,  6,  8,  11,  13,  or  6,  11,  13. 

68-69  While  a  particular  form  of  answer  should  not  be 
insisted  upon,  tlie  answers  given  should  be  corrected  if  they  are 
not  clear.  A  good  form  may  be  given  by  the  teacher,  as  in  a, 
page  68  :  "  If  I  have  a  dime  and  a  five-cent  piece  I  shall  have 
fifteen  cents,  and  I  shall  need  two  cents  more  to  have  seventeen 
cents  "  ;  and  in  h :  "  Two  boys  have  four  legs  and  three  dogs  have 
twelve  legs.  Two  boys  and  three  dogs  have  four  legs  and  twelve 
legs.  Four  legs  and  twelve  legs  are  sixteen  legs."  If  the  children 
can  write  without  a  copy,  it  may  be  good  exercise  in  written 
language  for  them  to  write  the  answers  as  they  have  been  ac- 
customed to  recite  them  in  the  class. 

70  In  teaching  to  write  each  new  number,  pay  particular  atten- 
tion to  the  expression  of  the  ten  and  the  units  as  shown  on  page 
16  of  the  Manual. 

7  3  These  should  be  given  at  sight.  In  recitation  do  not  permit 
the  children  to  read  the  question  first  and  then  give  the  answer  by 
repeating  a  part  of  the  question.  The  answers  should  be  given  in 
entire  statements,  Avith  no  pause  before  the  answer.  Occasionally 
the  children  might  give  the  answer  quickly  to  each  question  by  a 
single  word. 

73  Sight  exercises  for  drill.  Let  the  children  give  results 
promptly,  thus  :  7,  10,  8,  5,  etc. 

74  Let  the  answers  to  these  problems  be  given  in  entire  sen- 
tences. The  solution  of  some  of  them  may  have  to  be  pictured 
before  they  are  understood  by  the  children. 


I.  78]  teachers'  manual.  21 

78  Show  by  bundles  of  sticks  the  expression  of  two  tens  and 
no  units. 

81  A  portion  of  this  page  only  may  be  given  at  a  time.  Observe 
the  order  of  representation.  Keview  frequently  what  has  been 
taught. 

8-4-80  The  problems  of  these  pages  will  suggest  a  kind  of 
work  that  may  be  done  in  connection  with  nature  lessons  and 
language.  Lead  the  children  to  give  tlie  ansAvers  in  entire  sen- 
tences. Thus,  in  26,  page  84,  the  answers  may  be  :  (o)  3 
pansies  have  3  times  5  petals.  3  times  5  petals  are  15  petals. 
(J>)  1  pay  G  times  2  cents  for  6  two-cent  postage  stamps.  G  times 
2  cents  are  12  cents.  I  pay  5  cents  for  the  paper.  I  pay  12  cents 
and  5  cents  for  the  stamps  and  paper.  12  cents  and  5  cents  are 
17  cents. 

Let  the  statements  be  made  by  the  children  in  reply  to  questions. 
Gradually  they  will  be  able  to  give  the  answers  in  proper  form 
without  assistance. 


SECTION   IV. 

NOTES   FOR   BOOK   NUMBER   TWO. 

For  a  brief  statement  of  the  purposes  of  this  book  and  for  some 
general  directions  see  the  Note  to  Teachers  given  in  Book  No.  2. 
It  may  be  said  further,  that  objects  should  be  used  in  teaching 
every  new  fact  or  process.  The  illustrative  work  here  given  is 
intended  to  supplement  and  not  to  take  the  place  of  such  teaching. 
The  objects  needed  will  be  pasteboard  squares  and  sticks  for 
counters,  common  weights  and  measures,  and  toy  money.  The 
kind  of  illustrative  blackboard  work  which  may  be  done  by 
teachers  will  appear  in  the  notes. 

After  the  various  operations  have  been  taught,  much  drill  in- 
volving a  repetition  of  the  mental  act  in  learning  them  will  be 
found  necessary.  There  is  given,  accordingly,  in  this  book  a  great 
variety  and  number  of  drill  problems.     For  variety  of  class-work 


22  GEADED   ARITHMETIC.  [II.  1 

it  may  be  well  to  give  exercises  from  the  blackboard  from  time  to 
time,  varying  with  the  needs  and  progress  of  the  children,  but  for 
seat-work  it  is  believed  that  the  given  exercises  will  furnish 
abundant  material. 

1—11  These  problems  are  a  review  of  work  given  in  Book 
No.  I.  If  the  children  cannot  perform  them  with  some  degree  of 
promptness,  it  is  advised  that  needed  portions  of  the  previous  book 
be  taken  up.  Possibly  there  will  be  needed  only  some  oral  or 
blackboard  drill.  It  may  be  well  to  place  upon  the  board  the 
following  for  drill  in  addition : 

111111111 
123456789 

22222222 
23456789 

3     3     3     3     3     3     3 

3  4     5     6     7     8     9 

4  4     4     4     4     4 

4  5     6     7     8     9 

5  5     5     5     5 

5  6     7     8     9 

6  6     6     6 

6  7     8     9 

7  7  7  These  figures  represent  all  pos- 

7  3  9  sible  pairs  of  units  that  can  be 
~  ~  ~                             made.     There  is  no  combination 

8  8  in  simple  addition  that  may  not 
8  9  be  referred   to  them.     It  is    for 

this  reason  that  a  thorough 
knowledge  of  these  combinations 
should  be  had. 


II.  12]  teachers'  manual.  23 

13—13  If  necessary,  before  giving  these  lessons  from  the 
book,  teach  the  writing  of  numbers  to  20  with  the  aid  of  sticks 
as  suggested  on  page  17  of  the  ManuaL  Further  drill  similar  to 
that  given  on  page  12  may  be  given. 

14-19  The  teaching  of  numbers  to  100  by  means  of  sticks 
should  precede  and  accompany  these  exercises.  The  purpose  of 
such  teaching  is  to  give  the  children  a  good  foundation  knowledge 
of  numbers  and  their  expression.  As  before,  every  ten  of  sticks 
should  be  bound  into  a  bundle,  and  the  number  of  such  bundles  in 
a  number  should  be  called  so  many  tens,  while  the  single  sticks 
should  be  called  ones  or  units.  For  example,  if  twenty-six  is  the 
number  to  be  taught,  let  the  teacher  count  out  ten  and  place  a 
rubber  band  about  it,  proceeding  as  follows  :  "  How  many  sticks 
have  I  here  ?  We  will  call  it  a  what  ?  Yes,  a  ten.  Let  us  count 
out  ten  more.  How  many  tens  have  I  now  in  my  hand  ?  How 
many  more  sticks  have  I  ?  How  many  sticks  in  all  ?  Two  tens 
of  sticks  and  six  sticks  are  how  many  sticks  ?  Now  let  us  write 
this  number  on  the  board  (placing  the  two  bundles  together  and 
six  sticks  together  on  the  ledge).  Who  will  place  above  the  tens 
bundles  the  figure  which  tells  how  many  tens  ?  Who  will  place 
above  the  single  sticks  the  figure  which  tells  the  number  of  ones 
or  units  ?  Who  will  read  the  whole  number  ?  "  The  same  method 
should  be  pursued  with  other  numbers  in  the  twenties  and  tliirties, 
and  then  the  children  should  be  given  sticks  and  bands  to  count 
out  and  express  given  numbers  as  far  as  forty.  Tlie  children  are 
now  ready  to  do  the  work  called  for  on  page  14.  The  same  course 
should  be  followed  in  teaching  the  numbers  to  100.  Several  days 
may  be  profitably  spent  upon  this  work,  the  book  beiug  used  for 
review  in  the  recitation  and  for  busy-work.  Lead  the  children  to 
draw  the  squares  very  carefully  with  a  ruler  or  other  straight 
edge. 

20    A  review  which  should  be  performed  without  objects. 

31—26  Observe  carefully  the  order  of  these  problems,  taking 
u])  new  Avork  only  when  the  old  is  well  understood.  Lead  the 
children  to  perform  the  work  of  addition  and  subtraction  first  with 


24  GRADED    ARITHMETIC.  [II.  27 

sticks,  afterwards  by  means  of  drawn  squares,  and  finally  without 
the  aid  of  objects.  Do  not  proceed  too  quickly  to  the  abstract 
work  because  the  children  can  give  answers  to  the  questions.  One 
important  purpose  of  using  objects  at  this  stage  is  to  give  a  good 
foundation  for  subsequent  work. 

The  problems  on  page  23  are  intended  for  sight-work,  but  it  may 
be  well  to  have  the  children  repeat  the  problem  first  before  giving 
the  answer  ;  thus  :  Twenty  and  ten  are  thirty,  and  four  are  thirty- 
four  ;  thirty  and  twenty  are  fifty,  and  three  are  fifty -three,  etc. 
Fifty-six  less  twenty  are  thirty-six,  etc.  If  tlie  children  hesitate 
in  giving  answers,  go  back  one  step  ;  for  example,  if  they  cannot 
give  an  answer  at  once  to  the  question  53  -j-  40,  give  jDroblems  in 
the  addition  of  tens  alone,  30  +  20,  40  +  30,  50  +  40,  etc.  The 
same  course  should  be  pursued  in  subtraction. 

The  questions  on  page  24  should  also  be  repeated  thus  :  Five  and 
two  are  seven,  fifteen  and  tAvo  are  seventeen,  etc.  Adding  so  as  to 
make  even  tens,  and  subtracting  so  as  to  break  up  even  tens, 
should  be  fully  shown  by  objects  and  drawings.  Possibly  similar 
work  will  have  to  be  given  by  drill  from  the  board. 

27  —  28  The  children  should  read  these  problems  silently,  and 
repeat  a  portion  of  them  in  the  answer  ;  thus  in  4,  page  27,  the 
answer  might  be  :  "  If  I  sleep  ten  hours  of  the  day  I  shall  be 
awake  twenty-four  hours  less  ten  hours.  Twenty-four  hours  less 
ten  hours  are  fourteen  hours."  Of  course  other  forms  of  answers 
should  be  permitted. 

29  —  31  These  examples  can  be  performed  by  the  children 
without  objects,  and  they  will  not  be  found  very  difficult ;  but  it 
will  be  far  better  to  use  the  sticks  for  a  day  or  two  u})on  similar 
examples,  both  for  the  expression  of  the  numbers  and  for  the 
process  of  adding  or  sul)trafting.  This  will  be  especially  neces- 
sary in  subtraction.  Thus  to  take  GO  from  84,  the  children  take 
8  tens  of  sticks  and  4  sticks,  and  express  that  number  in  writing 
.as  84.  They  are  then  asked  to  take  6  tens  of  sticks  or  ()0  sticks 
from  the  iiuiiil)('r.  They  do  tliis  and  find  24  sticks  remaining. 
This  process  they  represent  by  figures,  writing  one  number  under 


II.  32]  teachers'  manual.  25 

the  other,  drawing  a  line  under  the  lower  number,  and  writing  the 
remainder  below.  All  this  is  a  good  preparation  not  only  for  the 
problems  of  these  three  pages  but  also  for  the  problems  which 
follow. 

33  —  35  Here  is  taken  a  new  and  important  step  in  addition 
and  subtraction  —  the  adding  of  units  whose  sum  is  more  than  ten, 
and  the  su.btracting  of  units  from  a  ten  and  units,  the  iinits  of  tlie 
whole  being  less  than  the  units  of  the  part.  The  children  have 
learned  to  perform  these  operations  to  twenty,  and  now  they  are  to 
apply  the  knowledge  thus  gained  to  larger  numbers.  Sticks  should 
be  used  as  before,  first  by  the  addition  of  2  to  9,  making  1  ten  and 
1  unit,  and  afterwards  by  the  addition  of  the  same  number  to  19, 
29,  39,  etc.  Let  the  children  stop  in  each  case  to  put  the  band 
about  the  new  bundle  of  ten,  and  express  in  figures  what  they  have 
done.  The  same  course  should  be  taken  in  subtraction.  Tlie  book 
may  now  be  taken  for  the  solution  of  problems  by  the  aid  of  the 
cuts.  The  drawing  of  the  squares  in  the  manner  indicated  on 
pages  32  and  34  will  be  good  busy-work  for  the  children.  The 
answers  may  first  be  given  by  repeating  the  numbers  to  be  added 
or  subtracted,  and  afterAvards  at  sight  by  giving  the  result  alone  ; 
thus,  in  3,  page  33  :  Eleven  less  two  are  nine,  twenty-one  less  two 
are  nineteen,  etc.  Nine,  nineteen,  twenty-nine,  etc.  The  children 
will  be  able  by  repeated  drill  to  add  and  subtract  by  twos  and 
threes  very  rapidly. 

3(>  — 40  There  is  no  new  principle  involved  in  the  problems  of 
these  pages,  and  the  same  course  should  be  followed  in  teaching  and 
drilling  which  was  pursued  in  the  previous  four  pages.  Do  not 
neglect  to  use  the  ol)jects  and  drawings  Avhenever  a  new  nuMiber 
is  to  be  added  or  subtracted.  A  repetition  of  the  mental  act  which 
accompanies  the  work  with  objects  is  as  necessary  as  the  repetition 
of  the  mental  act  in  recalling  former  impressions.  Very  much 
diill  is  needed,  both  with  and  Avithout  objects,  to  iix  the  c(iHd)i!i;;- 
tions  in  tlie  minds  of  the  children,  so  that  they  Avill  be  given  with 
perfect  accuracy  and  promptness.  Occasional  drill-work  on  tlie 
board  may  be  found  useful.      Fur  busy-work,  lead  the  chiUlren  to 


26  GRADED    ARITHMETIC.  [II.  47 

write  out  problems  in  the  addition  and  subtraction  of  5,  6,  etc.,  in 
the  manner  indicated  on  page  37.  Tliis  will  aid  them  in  future 
written  work. 

47  A  simple  extension  of  the  process  already  learned  is  required 
here.  Use  objects  and  drawings  of  lines,  dots,  or  squares.  In  the 
cut  on  this  page  let  the  children  place  a  card  over  the  squares  in 
such  a  way  as  to  show  2  threes,  3  threes,  4  threes,  etc.  Read  in  6, 
Two  multiplied  by  four,  or  four  times  two. 

48  The  getting  of  fractional  parts  of  numbers  by  the  aid  of 
objects  and  drawings  is  very  interesting  and  useful  work  for  the 
children,  and  should  be  frequently  done.  This  objective  work 
should  be  immediately  followed  by  a  repetition  of  the  solution  of  the 
problem  without  the  objects.  For  example,  after  the  children  have 
learned  that  i  of  12  dots  is  4  dots,  and  that  §  of  12  squares  are  8 
squares,  they  should  be  asked  the  questions,  ''  What  is  one-third  of 
twelve  ?  two-thirds  of  twelve  ?  " 

The  division  of  numbers  by  numbers  is  the  reverse  of  multiplica- 
tion, and  may  be  taught  directly  in  conpection  with  it.  After 
counting  out  24  sticks,  have  the  children  separate  them  into 
groups  of  3.  Then  ask,  '■'■  How  many  threes  of  sticks  are  there 
in  twenty-four  sticks  ? "  or,  "  Three  sticks  in  twenty-four  sticks, 
how  many  times  ? "  The  same  may  be  done  with  the  drawn 
squares,  letting  the  column  or  line  of  squares  represent  the 
divisor,  and  the  uncovered  squares  the  dividend. 

49  —  51  Partition  and  division  by  4  are  to  be  taught  and  drilled 
upon  as  was  indicated  for  the  same  processes  by  3.  Before  the 
solution  of  applied  problems,  the  children  should  be  able  to  give 
with  a  considerable  degree  of  promptness  answers  to  all  examples 
on  these  pages. 

53  —  54  The  most  difficult  of  these  problems  should  be  taught 
with  objects  in  the  recitation,  but  in  teaching  them  be  careful  not 
to  (h)  anything  for  tlie  children  which  tliey  can  do  themselves. 
For  examph;,  in  8,  l)age  52,  you  say  first,  "We  will  let  the  sticks 
represent  eggs,  and  you  may  count  out  one  dozen.  How  many  eggs 
have  you,  "NN'illie  V     How  many  have    you,  Mary  ?     Now,  if   my 


II.  55]  teachers'  manual.  27 

hens  lay  four  eggs  every  diiy,  liuw  many  days  will  it  take  them  to 
lay  the  dozen."  Do  not  disturb  the  children  in  their  thinking, 
until  you  see  tliat  the  process  is  not  clear.  If  it  is  not  clear  say  : 
"If  they  la}'  four  eggs  iu  a,  day  covnit  out  how  many  they  would 
lay  in  tw^o  days.  oSTow  see  if  you  can  find  how  many  days  it  will 
take  the  hens  to  lay  a  dozen."  Probably  uo  further  help  will  be 
needed.  Tlien  proceed  with  the  larger  number  recpiired,  varying 
the  number,  and  calling  upon  children  singly.  After  the  most 
difficult  of  the  problems  are  taught,  let  the  children  "  picture " 
them,  and  others  which  have  not  been  taught.  Thus,  in  the  above 
problem,  the  children  could  be  led  easily  to  illustrate  it  as  follows  : 

1  (lav.  1  clay.  1  day. 


and  to  write  below  the  pictured  solution  :  "  It  will  take  3  days  for 
the  hens  to  lay  a  dozen  eggs  if  they  lay  4  eggs  a  day."  If  it  is 
thought  best  to  have  the  representation  of  objects  more  realistic, 
ovals  instead  of  lines  could  be  drawn. 

After  the  problems  have  been  taught  and  pictured,  the  children 
should  be  ready  to  solve  them  without  aids,  and  give  the  answers 
in  entire  sentences,  as  in  5,  page  53  :  "  From  the  fourth  to  the 
twenty-second  of  June  there  are  eighteen  days.  In  eighteen  days 
there  are  two  weeks  and  four  days." 

55  — 5G  Xot  much  time  is  needed  for  the  multiplication  and 
division  by  five,  and  yet  it  is  not  well  to  neglect  any  part  of  the 
objective  work.  The  addition  and  subtraction  work  indicated  on 
page  55  may  be  extended  if  it  is  found  that  the  children  are  losing 
their  hold  of  such  work. 

57  It  will  probably  not  be  necessary  to  teach  any  of  these 
problems.  If  not,  give  them  to  the  children  as  they  are,  to  work 
out  with  and  without  drawings. 

58  —  59  In  teaching  and  drilling,  dwell  particularly  upon  those 
examples  which  are  most  difficult  to  remember,  as  for  example, 
6  X  7,  6  X  9,  42  -^  6,  54  -i-  6.    For  review  in  addition  and  subtraction 


28  GRADED   ARITHMETIC.  [II.  60 

as  well  as  in  division,  practice  considerably  upon  such  examples  as 
10  to  13  and  16,  page  59.  This  may  be  done  by  giving  a  number, 
as  38,  which  the  cliildren  are  exj^ected  to  divide  by  6  or  5.  The 
answer  would  be,  "  There  are  6  sixes  and  2  in  38,"  or,  "  6  is  con- 
tained in  38,  6  times  and  2  remainder." 

CO  If  the  previous  work  lias  been  thoroughly  done,  the  children 
will  find  little  difficulty  in  performing  these  problems  according  to 
the  model  shown  in  1. 

61-63  7  X  6,  7  X  8,  and  7X9  are  likely  to  give  the  children 
most  trouble  to  remember.  Drill  upon  these  combinations,  there- 
fore, should  be  most  persistent.  The  other  multiplications  by  4 
and  6  have  been  learned  previously,  Avith  the  factors  inverted. 
The  last  four  numbers  of  page  62  might  be  extended  considerably 
for  oral  drill. 

63  A  little  assistance  may  have  to  be  given  to  the  children 
before  they  can  work  out  the  exercises  as  indicated.  Do  not  hasten 
them  with  this  work,  and  when  any  child  finds  an  object  too  difficult 
to  draw,  let  him  use  marks  or  dots.  Chairs,  horses,  and  children  will 
probably  be  too  difficult  for  children  to  draw,  and  any  attempt  to 
do  it  might  divert  their  attention  from  the  numerical  operations. 

64  Dwell  with  special  emphasis  upon  the  most  difficult  com- 
binations in  both  multiplication  and  division.  Frequent  additions 
and  subtractions  by  eights,  beginning  with  8  and  9G,  will  lielp  to 
fix  the  tables  in  the  minds  of  the  children. 

65  These  problems  should  be  performed  first  by  picturing  as 
called  for,  and  afterwards  without  aids  of  any  kind. 

66  See  suggestions  for  page  64. 

67  This  is  a  review  of  what  may  profitably  be  done  in  various 
stages  of  the  cliildren's  progress.  Let  the  numbers  of  the  table 
which  are  most  difficult  to  learn  be  written  in  heavier  lines  than 
the  others. 

68  — (>9  Review  drill  exercises  for  busy-work  and  for  recita- 
tion. 

70  To  be  performed  without  aids  if  possible.  It  may  be  neces- 
sary to  give  some  assistance  in  15. 


II.  71]  teachers'  manual.  29 

71—73  Weights  and  measures  new  to  tlie  cliildren  sliould  be 
taught  with  tlie  objects.  The  simpler  problems  also  should  be 
first  taught  with  objects.  Afterwards,  pictures  representing  the 
weights  and  measures  may  be  used.  Squares  or  oblongs  of  ap- 
proximately correct  relative  size  would  suffice  for  this  purpose. 
For  example,  in  11,  a  square  whose  edge  is  1  in.  could  represent 
the  bushel,  marked  off  in  four  equal  parts  to  represent  pecks. 
Opposite  the  spaces  the  prices  could  be  placed,  and  from  them  the 
required  answers  given.  Problems  involving  the  use  of  pounds 
and  dozens  also  can  be  illustrated  in  the  same  way.  The  children 
if  permitted  will  give  some  very  interesting  graphic  solutions  of 
these  problems.  When  the  children  have  a  clear  idea  of  the  pro- 
cesses by  the  aid  of  objects  and  drawings,  they  should  be  expected 
to  solve  the  problems  promptly  without  aid  of  any  kind. 

74  When  the  children  know  thoroughly  and  can  tell  at  sight 
all  combinations  to  100,  using  any  number  for  the  adding,  sub- 
tracting, multiplying,  or  dividing  number  to  10,  they  are  ready  for 
the  work  of  this  section,  which  includes  the  addition  and  subtrac- 
tion of  any  number  to  100,  and  the  multiplication  and  division  of 
numbers  to  100  by  numbers  to  20.  To  accomplish  this  it  will  be 
necessary  to  follow  slowly  the  order  indicated,  and  to  give  frequent 
reviews. 

75  —  82  If  the  work  is  found  too  difficult  at  any  point,  observe 
the  preceding  steps.  For  example,  if  the  children  cannot  perform 
readily  the  work  on  page  76,  let  them  first  add  by  tens  and  units 
separately  ;  thus,  in  10  :  30  and  40  are  70,  and  9  are  79  ;  50  and 
30  are  80,  and  2  are  82,  etc.  After  a  rapid  review  of  this  kind 
they  will  be  ready  to  analyze  and  add  mentally.  Subtraction  will 
be  found  more  difficult  than  addition,  and  more  time  sliould  be 
spent  upon  it.  The  separation  of  the  number  to  be  subtracted  into 
tens  and  units  may  be  frequently  necessary ;  for  example,  in  2, 
page  81  :  75  less  30  are  45  less  9  are  36,  etc.  A  complete  rnastery 
of  the  subtraction  of  any  number  from  100  is  especially  desirable, 
since  such  subtraction  will  be  found  very  convenient  in  making 
change.     There  is  no  more  reason  for  adding  coins  piece  by  piece 


30  GRADED    ARITHMETIC.  [II.  83 

in  making  change  for  a  dollar  than  in  making  change  for  a  dime. 
One  good  method  of  drill  in  subtraction  is  to  place  a  number  upon 
the  blackboard,  and  give  orally  a  number  with  the  expectation  that 
the  children  will  name  anotlier  number,  which  added  to  the  given 
number  will  make  the  number  on  the  board.  For  example, 
the  teacher  writes  89  on  the  board,  and  says  :  Forty-six.  Pujnls. 
Forty-three.  T.  Twenty-four.  P.  Sixty-five,  etc.  Rapid  individ- 
ual work  of  this  kind  will  be  found  useful. 

83  —  84:  Tlie  previous  work  of  this  section  is  expected  to  be 
performed  witliout  the  aid  of  figures.  For  variety,  and  for  the 
purpose  of  keeping  fresh  in  the  children's  minds  the  decimal 
system  of  notation,  these  written  exercises  are  given.  If  the 
processes  required  are  not  fully  understood,  let  two  or  three  of  the 
examples  be  performed  with  sticks,  as  was  previously  shown.  Then 
let  the  children  add,  explaining  the  process  of  "carrying"  as  they 
add.  The  explanation  may  be  very  simple ;  thus,  in  36  :  9  units 
and  4  units  are  13  units,  equal  to  1  ten  and  3  units.  Write  3  units 
in  the  place  of  units.  1  ten  and  5  tens  and  3  tens  are  9  tens. 
Write  9  tens  in  the  place  of  tens.  Answer:  9  tens  and  3  units, 
or  93. 

85  —  86  An  almost  unlimited  number  of  examples  may  be 
given  in  connection  with  these  tables.  It  will  be  observed  that 
the  numbers  are  arranged  differently  in  the  three  tables,  and  that 
they  can  therefore  be  used  at  any  time  for  a  particular  purpose. 
For  example,  if  it  is  found  that  the  combination  8  +  -"^  ii^  connec- 
tion with  large  numbers  is  especially  hard  for  the  children  to 
remember,  drill  may  first  be  given  in  Table  A,  lines  s,  j^,  and  n. 
Add  8  to  numbers  in  line  p  as  far  as  i.  Add  18  as  far  as  g. 
Add  38  as  far  as  /.  Do  the  same  to  numbers  in  .s.  Subtract  5, 
15,  etc.,  from  line  n.  The  same  may  next  be  done  with  Table  B, 
lines  0,  p,  and  n. 

If  it  is  desired  to  drill  with  14  as  a  subtrahend,  the  order  of 
drill-work  would  be  :  In  Table  B,  subtract  14  from  numbers  in 
columns  c,  d,  e,  etc.  Subtract  14  from  numbers  in  lines  I,  m,  etc., 
beginning   with   c.     In    Table    C,   subtract   14   from   numbers   in 


II.  87]  TEACHERS*    MANUAL.  31 

columns  c,  d,  e,  etc.  Subtract  14  from  numbers  in  line  I  beginning 
with  e,  from  numbers  in  ///,  n,  etc. 

If  it  is  ilesired  to  add  or  subtract  a  number  from  numbers  whose 
tens  are  the  same  and  whose  units  are  unlike,  the  columns  in  tables 
B  and  C,  could  be  used  ;  and  if  it  is  desired  to  add  or  subtract  a 
number  from  numbers  whose  tens  are  unlike  and  whose  units  are 
the  same,  the  lines  of  A  and  B  could  be  used.  Other  uses  of  the 
tables  will  appear  for  multi})li('ation  and  division. 

For  methods  of  blackboard  drill,  see  Manual,  pages  33  and  34. 

87-l)t)  Ability  to  multiply  numbers  by  any  number  to  10,  and 
to  divide  by  numbers  to  20,  would  seem  to  be  of  sufficient  conven- 
ience in  everyday  life  to  give  some  attention  to  these  processes  in 
the  lower  grades.  Tlie  nse  of  the  drill  tables  on  the  two  preceding 
pages  may  help  to  sup|)lement  these  pages  in  giving  the  needed 
drill.  Place  ujion  the  board  other  drill  tables,  consisting  of  the 
multiples  to  100  of  all  numbers  as  far  as  20,  ■ —  one  with  the 
multiples  set  in  order,  and  another  with  the  multiples  not  in  any 
regular  order.  Drill  u})un  these  until  the  multiples  are  readily 
recognized.  There  is  no  reason  why  72  cannot  be  as  quickly 
recognized  to  be  a  multiple  of  18  as  of  9. 

91  Some  attention  to  the  form  of  answers  to  problems  should 
be  given,  with  the  understanding  always  that  a  correct  form  in 
itself  is  not  a,n  end,  but  only  a  means  of  showing  that  the  process 
of  thinking  is  correct.  First  be  sure  that  the  child  understands  a 
process,  and  then  lead  him  to  express  the  steps  of  the  process  in 
correct  language.  This  page  of  problems,  and  others  which  follow, 
are  supposed  to  furnish  models  of  simple  explanations.  To  fix 
some  of  these  forms  in  the  mind,  it  might  be  well  to  give  other 
numbers  than  those  stated  in  a  problem.  For  example,  in  9,  after 
the  problem  is  solved  as  given,  the  teacher  might  say,  "  What  will 
four  books  cost  ?  "  or,  "  Suppose  six  books  cost  eighteen  dollars, 
what  would  one  book  cost  ?  two  books  ?  "  etc.  If  the  thought  is 
not  clearly  expressed,  do  not  correct  the  form  of  answers  until 
it  is  clear  that  the  process  is  understood.  Objects  or  drawings 
should  frequently  be  used,  both  in  testing  the  child's  knowledge  of 
a  process,  and  in  teaching  him  the  process. 


32  GRADED   ARITHMETIC.  [II.  92 

92  Some  of  these  problems  may  have  to  be  ilhistrated  before 
the  statements  of  steps  and  answers  are  given,  but  let  the  children 
illustrate  them  if  they  can  without  assistance.  With  the  illustra- 
tions before  the  children,  the  teacher  may  ask,  as  in  5,  "  One  apple 
costs  what?"  "  Eight  apples  will  cost  how  many  times  two  cents  ?" 
Then  the  problem  can  be  solved  in  the  form  given,  to  be  followed 
by  the  use  of  other  numbers.  The  statement  of  steps  in  short 
sentences,  as  in  7  and  8,  will  be  found  useful  and  i)rofitable  as 
busy-work.  If  in  any  case  the  numbers  seem  too  large,  use  smaller 
ones  ;  as  in  3,  the  teacher  maj  say,  "  What  will  it  cost  at  that  rate 
to  get  two  collars  washed  ?  four  collars  ?  eight  collars  ?  four 
collars  and  two  cuffs  ?  "  etc. 

93  A  little  preliminary  questioning  or  teaching  may  be  ad- 
visable before  some  of  these  problems  are  given  ;  for  example, 
3  and  6.  Problems  similar  to  4  will  be  found  interesting  and 
profitable.  The  processes  involved  in  8  and  9  will  be  readily 
understood  if  similar  problems  with  smaller  numbers  are  first 
given. 

94  To  learn  once  for  all  the  number  of  days  in  each  month  is 
quite  important,  and  need  not  be  so  difficult  as  many  suppose. 
The  verse,  "  Thirty  days  hath  September,"  could  be  learned,  but  it 
would  be  better  to  learn  the  facts  that  four  of  the  months  have  30 
days,  that  one  has  28  in  all  years  except  leap  years,  and  that  all  the 
rest  have  31  days.  Erequent  applications  of  these  facts  in  little 
problems  will  fix  them  in  the  mind  so  that  they  will  not  be  for- 
gotten. Children  never  tire  of  measuring,  an-d  the  rod  line  which 
the  children  are  directed  to  make,  will  be  a  never-ending  source  of 
pleasure  and  profit. 

95  This  work  should  be  continued  until  the  children  have  a 
tolerably  accurate  idea  of  short  distances. 

90  lleduction,  first  with  the  aid  of  a  measure,  and  afterwards 
as  far  as  possible  with  no  aid,  will  be  found  an  excellent  prepara- 
tion for  subsequent  work.  Paper  of  different  sizes  should  be 
counted  out  in  sheets  and  quires,  and  some  simple  problems  given 
with  the  paper  in  sight  of  the  children. 


11.  J>7]  teachers'  imanual.  •  33 

97-104:  Most  of  the  facts  contained  in  these  problems  should 
have  been  taught  previously.  If  in  any  problem  there  is  a  fact  or 
process  quite  new  to  the  children,  it  shouhl  be  taught  objectively, 
the  rule  being  observed  that  nothing  sliould  be  told  the  child  which 
he  can  find  out  for  himself.  The  illustration  of  problems  by  the 
children  is  advised  whenever  the  processes  seem  too  difficult,  but 
the  same  or  similar  problems  should  afterwards  be  performed 
without  the  aid  of  objects  or  drawings. 


Blackboard  Drill.  —  For  the  sake  of  variety,  it  may  be  well 
sometimes  to  give  a  drill  in  adding,  subtracting,  multiplying,  and 
dividing  from  the  blackboard.  If  it  is  found,  for  example,  that 
the  pupils  are  slow  in  adding  and  subtracting,  put  on  tlie  board 
columns  beginning  with  only  ones  and  twos,  and  increasing ;  new 
numbers  to  be  added  or  subtracted  gradually.  Practice  should 
continue  until  as  great  facility  is  had  in  adding  and  subtracting  7, 
8,  and  9,  as  in  adding  and  subtracting  1,  2,  and  3.  The  columns 
will  appear  as  follows  : 


1 

3 

o 

3 

3 

4 

3 

4 

5 

3 

8 

G 

8 

9 

2 

1 

1 

o 

2 

3 

6 

3 

G 

8 

3 

8 

9 

6 

1 

2 

3 

1 

4 

5 

9 

9 

r^ 

i 

4 

r* 

1 

7 

8 

9 

2 

2 

4 

3 

5 

3 

4 

4 

5 

G 

9 

G 

3 

2 

3 

2 

4 

1 

9 

3 

9 

Ad 

G 

8 

o 

5 

4 

1 

1 

1 

9 

3 

4 

5 

5 

r- 
( 

1 

4 

1 

2 

3 

4 

4 

5 

G 

G 

•  > 

4 

8 

3 

8 

8 

2 

3 

4 

3 

2 

4 

2 

3 

5 

5 

G 

8 

7 

9 

1 

2 

2 

1 

5 

3 

3 

4 

4 

G 

3 

7 

9 

8 

2 

1 

3 

4 

1 

9 

6 

G 

7 

8 

G 

5 

9 

2 

3 

1 

3 

3 

r> 

G 

5 

3 

G 

5 

5 

Entirely  new  combinations  may  be  made  b}-  substituting  a  num- 
ber in  place  of  the  first  number  of  the  column  to  be  added,  or  by 
placing  a  new  number  below  or  above  the  column,  and  beginning 
with  that  number  to  add. 


34 


GRADED   ARITHMETIC. 


The  same  columns  may  be  used  for  subtraction  by  placing  any 
number  above  the  column,  from  which  successive  subtractions  are 
made.  Thus,  if  50  were  placed  above  the  first  column,  the  children 
would  be  led  to  say,  49,  47,  46,  44,  etc. 

By  the  same  method,  correctness  of  work  in  addition  may  be 
proved.  For  example,  if  tlie  pupils  in  adding  the  first  column 
have  an  answer  of  17,  they  can  begin  with  that  number  and  sub- 
tract successively  the  numbers  of  the  column.  If  in  subtracting 
the  last  number  the  remainder  is  nothing,  the  answer  is  presumed 
to  be  correct. 

For  drill  in  multiplication  and  division  the  same  columns  may 
be  used,  the  number  to  be  multiplied  or  divided  being  placed  above 
the  column. 

The  same  progressive  drill-work  may  be  given  by  placing  the 
numbers  in  a  circle,  as  follows  : 


The  pupils  could  add  in  either  direction  to  any  given  number, 
or  from  any  number  placed  in  the  centre,  subtractions  to  zero 
could  be  made.  Numbers  could  also  be  placed  in  the  centre  for 
multiplication  and  division. 


III.  1]  teachers'  manual.  36 


SECTION   V. 

NOTES   FOR   BOOK   NUMBER   THREE. 

By  far  the  greater  part  of  the  work  laid  out  in  this  book  is  with 
numbers  under  1000,  and  if  a  year  is  to  be  given  to  tlie  book,  at 
least  seven  months  may  Avell  be  spent  upon  this  part  of  it. 
Thoroughness  in  the  use  of  small  numbers  means  a  saving  of  time 
and  increased  power  to  apply  what  is  learned  in  the  higlier  grades. 

The  apparatus  needed  for  teaching  and  illustrating  these  exer- 
cises is  a  large  number  of  short  Avooden  sticks  with  rubber  bands, 
and  all  the  common  weights  and  measures,  such  as  the  foot  rule, 
yard  stick,  rod  line,  gill,  quart,  gallon,  peck,  bushel,  and  balance 
for  weighing. 

Eead  carefully  the  Note  to  Teachers  on  pages  iii  and  iv,  particu- 
larly what  is  said  in  the  paragraph  marked  2. 

1—11  These  exercises  are  a  partial  review  of  what  has  been 
given  in  Book  II.  Pupils  shoidd  be  able  to  perform  them  with 
facility  before  Section  II.  is  begun.  If  for  any  reason  there  is 
difhcidty  in  performing  or  in  understanding  any  of  the  exercises, 
they  should  be  tauglit  and  illustrated  as  previously  recommended. 
If  the  difficulty  lies  in  the  computations  involved  in  the  four 
fundamental  rules,  drill  is  needed  until  the  difficulty  is  removed. 
For  a  good  method  of  drill  upon  simple  computations  to  100  see 
Book  II.,  pages  85  and  86  ;  also  Manual,  pages  33  and  34. 

If  there  is  difficulty  in  making  the  applications,  the  use  of  small 
numbers  and  a  graphic  representation  of  the  processes  by  drawings 
are  advised. 

12—14  The  new  grouping  of  hundreds  to  make  the  thousand 
should  be  taught  by  objects  as  before,  and  if  the  objects  have 
previously  been  sticks  in  bundles  of  ten.  the  same  olijfcts  should 
now  be  used.  First  review  what  has  been  taught,  — 10  units  or 
ones  of  sticks  to  make  a  ten,  and  10  tens  of  sticks  to  make  a 


36  GRADED    ARITHMETIC.  [III.  15 

liundred.  Let  the  pupils  work  witli  these  tens,  and  units  until 
their  use  is  quite  familiar.  Lead  the  }mpils  to  put  into  bundles 
and  represent  2,  3,  4,  etc.,  hundreds,  varying  the  order  of  number- 
ing and  of  representing.  For  example,  the  teacher  puts  before  the 
pupils  piles  of  single  sticks  and  bundles  of  tens,  saying,  ''Show 
me  6  tens.  •  How  many  units  in  6  tens  ?  Who  can  write  that 
number  on  the  board  ?  ShoAv  me  seventy-six  ;  eighty-three  ;  sixty- 
four  ;  ninety-seven.  ShoAv  me  10  tens.  What  is  that  called  ?  How 
is  one  hundred  written  ?  Why  ciphers  in  the  units  and  tens 
places  ?  We  Avill  put  tlie  10  tens  together  in  a  bundle.  How 
many  hundreds  in  this  bundle  ?  Show  me  2  hundreds.  These 
together  are  called  two  hundred.  How  shall  Ave  write  the  number 
two  hundred  ?  Show  me  3  hundred ;  4  hundred.  AVho  will 
write  four  hundred  on  the  board  ?  ShoAv  me  1  hundred  and  2  tens. 
2  tens  is  called  what  ?  What  then  is  the  whole  number  called  ? 
How  will  you  write  1  hundred  and  2  tens,  or  one  hundred  twenty? 
Show  me  2  hundreds  and  3  tens.  What  shall  we  call  this  number  ? 
How  shall  we  write  it  ?  "  In  this  way  the  teacher  goes  on  teach- 
ing the  hundreds,  then  the  hundreds  and  tens,  and  finally  the 
hundreds,  tens,  and  units,  until  the  thousand  is  reached.  ISTow 
comes  the  drill  begun  on  page  12.  The  exercises  given  on  these 
pages  indicate  the  order  which  may  be  followed,  but  they  may  not 
be  sufficient  in  number  for  some  pupils. 

Teach  the  term  .min,  and  lead  the  puj)ils  to  use  it. 

15—16  These  exercises  are  supposed  to  be  performed  with  and 
without  the  aitl  of  objects  and  without  figures.  If  in  any  case  the 
answers  cannot  be  readily  given,  let  the  intermediate  ste})s  be 
given.  For  example,  in  11,  page  15,  let  the  pupils  say  :  660,  760, 
790  ;  380,  580,  630  ;  and  in  7,  page  10  :  850,  650,  41)0  ;  720,  420, 
380. 

1  7—1  S  These  exercises  should  be  performed  on  ])a])er  or  slate, 
first  with  sticks  and  afterwards  without  sticks.  Lead  the  jiupils 
to  pvit  together  the  sticks  in  each  case'  before  the  residt  is  written  ; 
thus,  in  19.  page  17,  the  ]mpils  should  first  ])nt  in  columns  the 
tens  and  units  of  sticks  to  be  added,  and  write  the  corresponding 


III.  17]  teachers'    IklANUAL.  37 

numbers  on  paper  or  slate.  They  then  put  together  the  3,  4,  and 
4  sticks.  10  of  these  they  should  put  into  a  ten  bundle  and  say, 
"  3  units  and  4  units  and  4  units  are  11  units,  equal  to  1  ten  and 
1  unit.  I  write  the  1  unit  in  the  place  of  units,  and  add  the  1  ten 
to  the  tens."  Then  putting  the  bundles  of  tens  together  they  say, 
''  1  ten  and  4  tens  and  2  tens  and  3  tens  are  10  tens,  equal  to 
1  hundred  and  0  tens.  I  write  the  1  hundred  and  0  tens  in  their 
places,  and  have  for  an  answer  101." 

Wlien  a  sufficient  amount  of  drill  with  the  sticks  has  been  given, 
the  pupils  should  place  in  columns  the  numbers  beginning  with  3, 
page  17,  and  proceed  in  order  until  all  exercises  on  these  pages  are 
performed. 

It  is  permitted  sometimes  for  convenience  in  "  proving  "         11 
answers  to  Avrite  the  "  carrying  "  figure  above  the  column         314 
in  which  the  addition  is  made  ;  but  it  is  not  well  for  pu})ils,         151 
either    in   addition    or    in  multiplication,  to  depend  upon         208 
writing  down  the  carrying  figure  j  thus,  37,  page  17  :  733 

314 
151 

A  good  way  of  preserving  the  carrying  figure  for  proof,  268 
is  to  write  the  result  of  each  addition,  thus  :  13 

12 
This  is  found  especially  useful  in  adding  long  columns.  0 

733 

Do  not  insist  upon  formal  and  elaborate  "  explanations."  It  is 
enough  for  the  pupils  to  express  what  they  recall  of  the  process  of 
uniting  together  ten  of  one  denomination  to  make  one  of  the  next 
higher.  For  example,  in  1,  page  18,  the  pupils  might  be  led  to 
say :  "2  units  and  7  units  and  6  units  are  15  units,  equal  to  1  ten 
and  5  units.  I  write  the  5  units  and  add  the  1  ten  with  the  tens. 
1  and  4  and  1  and  8  tens  are  14  tens,  e(|ual  to  1  hundred  and  4 
tens.  I  write  the  4  tens  and  add  the  1  hundred  with  the  hundreds. 
1  and  1  nnd  2  and  3  hiindreds  are  7  .hundreds,  which  I  write. 
Answer:  745."'     Oi  course  this  form  or  any  set  form  should  not  be 


38  GRADED    ARITHMETIC.  [III.  19 

put  before  them  as  a  model.  What  is  desired  of  them  is  simply  to 
recall  impressions  which  were  made  in  the  objective  work,  and  to 
express  what  they  think. 

19   Answers:   1    707.       2    861.       3    718.       4    841.       5    690. 

6  781.  7  765.  8  807.  9  795.  10  998.  11  867.  12  835. 
13  971.       14  840.       15  815.       16  854. 

A  large  number  of  drill  examples  may  be  made  by  asking  the 
pupils  to  place  given  numbers  above  or  below  in  the  columns,  or 
the  teacher  could  dictate  to  the  pupils  the  numbers  here  given  and 
one  other  number.  The  answers  could  be  found  by  adding  the  extra 
number  to  the  answers  above  given.  In  adding  aloud,  permit  only 
results  to  be  given,  and  lead  the  children  to  add  by  pairs.  If 
difficulty  is  found  in  this,  put  pairs  of  numbers  on  tlie  board  for 

them  to  add  quickly  at  sight  as  follows  :       ^       1 Q      i  J      ®^^-     ^^ 

will  not  be  difficult  after  some  practice  to  add  in  the  manner 
desired  ;  thus,  in  1,  the  answers  should  be  15,  26,  39,  46,  etc. 

30-Jil  These  exercises  should  be  performed  first  with  objects, 
the  pupils  being  led  to  use  the  sticks  as  indicated  before  the  results 
are  written  ;  thus,  in  12,  page  20,  tlie  pupils  place  before  tliem  6 
bundles  of  hundreds  and  7  bundles  of  tens,  and  after  writing  on 
paper  or  slate  670  above  286,  say,  "  I  wish  to  take  286  from  670." 
Having  no  units  they  take  a  ten-bundle,  and  after  seimrating  into 
units  and  taking  from  them  6  sticks,  say,  "  6  units  from  10  units 
are  4  units.     I  write  4  in  the  units  place.     I  took  1  ten  from  the 

7  tens,  and  there  are  left  6  tens.  I  cannot  take  8  tens  from  6  tens, 
so  I  take  one  of  the  hundreds."  This  he  does,  and  after  resolving 
it  into  tens  and  placing  them  with  the  6  tens,  says,  "  1  hundred 
equals  10  tens.  10  tens  and  6  tens  equal  16  tens."  He  then  goes 
on  with  the  subtraction,  writing  after  each  result  is  obtained,  and 
telling  what  is  done. 

As  in  addition,  the  statements  should  be  a  simj^le  and  natural 
expression  of  the  pui)il's  thought. 

At  this  point  teacli  the  terms  minuend,  suhtrdJiend,  and  remainder, 
and  lead  the  pupils  to  use  them  in  reading  tlie  exercises  and  answers. 


III.  22]  teachers'  manual.  39 

22  A}mvers:  1  591.  2  861.  3  961.  4  748.  5  976. 
6  900.  7  717.  8  803.  9  OO;?.  10  752.  11  957.  12  979. 
13  759.  14  1027.  15  866.  16  857.  17  9S8.  18  778. 
19  71-1     495     603.       20  307     350     159.       21  270. 

This  drill  table  may  he  used  in  supplying  a  large  number  of 
examples  in  addition  and  subtraction.  For  example,  the  pupils 
may  be  told  to  add  ba  from  A  to  K  or  from  I)  to  P,  or  the  Sailie 
work  may  be  dictated  to  the  pupils.  Other  columns  as  cb,  dc,  etc^, 
could  be  given  in  the  same'  way.  Practice  also  in  adding  by  lines 
could  be  given  from  the  table.  Other  columns  than  those  given 
for  subtraction  could  be  given  as  deb,  edc,  etc. 

23  A7iswei-s:  1  509.  2  216  492  307  139  219.  3  176 
317  252  517  187  186.  4  292  704  237  179  178  124. 
5  616  332  174  203  288  381.  6  206  337  186  291  144 
69.  7  278  184  289  249  336  186.  8  303  163  226 
340  203  197.  9  607  583  83  387  626  287.  10  963  cd. 
11  217  sheep.  12  625  hours.  13  71  highest  grades  176  lowest 
"Trades. 


& 


2-4    Answers:    1  226     194    292    393    398.       2  643  188    419 

361     206.       3  60     83     294     147     139.       4  39     280  361     335 

228.        5    348     266     719     304     422.        6    593     670  206     606 

445     101.        7  157.        8  268.        9   508.        10   319.  11   342. 

12  287.        13  306.       14  302.        15  305.        16  276.  17  321. 

18  244.       19  290.        20  372.        21  612.        22  426.  23  104 
72.       24  124     120.       25  164     82. 

From  this  table  a  large  amount  of  drill-work  may  be  given,  the 
pu})ils  taking  different  combinations. 

25  Answers:  1  873  bu.  corn  1026  Im.  wheat.  2  388  pear  trees. 
3  742  mi.  4  195  ft.  5  930  papers.  6  674  papers.  8  284  yrs. 
1492,  Dis.     1620,  Tilgrims     1776,  Dec.  of  Independence. 

Some  of  these  problems  suggest  a  kind  of  work  that  may  be 
given  from  statistics  found  in  the  latest  almanac. 


40  GRADED   ARITHMETIC.  [III.  26 

36    Answers:    1    365  d.       2    $396.        3    $65.        4    837  lb. 

5  1091.       6  350  mi. 

The  making  of  original  problems  is  good  practice  in  language. 
Useful  local  statistics  may  be  used  profitably  for  this  purpose. 

37—29  Observe  the  order  of  work  here  required,  and  do  not 
abandon  the  objects  because  pupils  can  find  the  correct  answers 
without  them. 

30  Answers:   1  99.         2  884.        3  604.  4  424.  5  624. 

6  798.  7  872.  8  590.  9  678.  10  720.  11  780.  12  840. 
13  805.  14  905.  15  928.  16  815.  17  748.  18  984. 
19  960.  20  772.  21  976.  22  819.  23  776.  24  897. 
25  936.  26  1088.  27  385.  28  978.  29  021.  30  820. 
31  915.  32  988.  33  588.  34  774.  35  1050.  36  995. 
37  861.  38  880.  39  872.  40  948.  41  957.  42  804. 
43  649.  44  963.  45  1062.  46  1188.  47  1228.  48  567. 
49  910. 

Let  work  with  objects  accompany  the  description  of  the  process 
and  precede  the  writing  of  results  ;  thus,  in  17,  the  pupils  should 
have  four  groups  of  sticks  to  put  together,  each  group  to  have  1 
hundred-bundle,  8  ten-bundles,  and  7  units.  In  putting  together 
the  single  sticks  or  units  the  pupils  should  say  :  "4  times  7  units 
are  28  units  ";  and  then  in  putting  the  twenty  into  two  bundles  of 
tens  should  add  :  "  28  units  equal  2  tens  and  8  units.  I  write 
the  8  units  in  the  place  of  units,  and  add  the  2  tens  to  tlie  next 
product."  Then  putting  together  the  tens  the  pupils  say  :  "  4  times 
8  tens  are  32  tens  plus  2  tens  are  34  tens,"  etc. 

Teach  the  terms  multiplicand,  multijdier,  and  product,  and  lead 
the  pupils  to  use  the  terms  in  reading  and  explaining  problems. 

31  Answers:  1  a  1044 ;  Z.  1216 ;  c  768  ;  fZ  984  ;  c  936. 
2  a  406  ;  6  672  ;  c  525  ;  d  574  ;  e  679.  3  a  783  ;  h  oi^5  ; 
c  828  ;  d  342  ;  e  657.  4  a  1296  ;  h  1496  ;  c  864  ;  d  1016  ; 
e  1472.  5  a  616  ;  b  528  ;  c  836  ;  c^  352  ;  e  737.  6  a  445 
534  623  712  801  890 ;  h  380  456  532  608  684  760 ; 
c  265     318     371     424     477     530 ;     d  460     552     644     736    828 


III.  32]         teachers'  ]MANUAL.  41 

920.  7  a  130  15G  182  208  234  2G0 ;  b  445  534  623 
712  801  890 ;  c  385  462  539  616  693  770 ;  d  465  558 
657  744  837  930.  8  a  410  492  574  656  738  820; 
h  465  558  657  744  837  930 ;  c  335  402  469  536  603 
670 ;  d  425  510  595  680  765  850.  9  a  385  462  539 
6J  6  693  770 ;  h  465  558  657  744  837  930 ;  c  430  516 
602  688  774  860;  d  135  162  189  216  243  270. 
10  a  380  456  532  608  684  760;  h  240  288  336  384 
432  480;  c  460  552  644  736  828  920;  d  380  456 
532  608  684  760.  11  a  385  462  539  616  693  770; 
b  340  408  476  544  612  680 ;  c  285  342  399  456  513 
570 ;  d  210  252  294  336  378  420.  12  a  445  534  623 
712  801  890;  b  380  456  532  608  684  760;  c  465 
558  657  744  837  930;  d  420  504  588  672  756  840. 
13  620  570  654  566  876  732  1026.  14  a  636;  b  768 
c  990  ;  d  1092  ;  e  1134  ;  /  1182.  15  a  609  ;  b  672  ;  c  518 
d  266  ;  e  574  ;  /  679.  16  a  448  ;  i  384  ;  c  576  ;  d  744 
e  520  ;  /  768.  17  a  513  ;  ^^  783  ;  c  324 ;  d  738  ;  e  882 
/774.  18  a  539;  ^.  726 ;  c  913;  (Z  836 ;  e  638 ;  /  737. 
19  70  41  51. 

33  Ansxvers:  1  a  787 ;  ^^  808 ;  c  1064.  2  a  964;  b  1078; 
c   1023.   3  a  992  ;  b   1067  ;  c   988.   4  «  288  ;  b   207  ;  c   463. 

5  a  150 ;  ^»  501 ;  c   688.   6  a  118 ;  Z;  146  ;  c  574. 

33  The  exercises  of  this  page  should  be  performed  with  and 
without  the  use  of  objects.  The  statement  of  steps  and  process 
given  should  be  the  same  in  one  case  as  in  the  other. 

34  Ajiswers:    1  736.        2  676.        3  594.        4  852.        5  812. 

6  645.  7  525.  8  946.  9  720.  10  756.  11  546.  12  931. 
13  782.  14  702.  15  528.  16  432.  17  1157.  18  1358. 
19  646.  20  1288.  21  1548.  22  931.  23  885.  24  741. 
25  1472.  26  1144.  27  425.  28  1148.  29  836.  30  1170. 
31  1066.  32  688.  33  476.  34  1022.  35  840.  36  a  1064; 
6  700  ;    c  1022  ;    c^  728  ;    e  1204  ;    /  1008.       37  a  795  ;    b  915  } 


42  GRADED    ARITHMETIC.  [III.  35 

c  1125  ;  d  1035  ;  e  855  ;  /  570.  38  a  464  ;  b  592  ;  c  832  ; 
0^  624  ;  e  720  ;  f  288.  39  a  663  ;  Z^  765  ;  c  323  ;  (/  408  ; 
e  527 ;    /  289.       40  1246     1035     1343.       41  1314     1273     910. 

35    Answers:    1  1134     300      235     1186.        2  623  21     254 

343.       3   552.       4   884.       5  837.       6   782.       7   770.  8    768. 

9  902.  10  864.  11  918.  12  903.  13  870.  14  832. 
15  980.  16  836.  17  936.  18  848.  19  806.  20  870. 
21  966.  22  621.  23  638.  24  700.  25  1054.  26  970. 
27  1073.       28  840.       29  882.       30  864. 

These  should  be  performed  without  objects,  the  pupils  giving  in 
clear  form  statements  of  process. 

The  objective  work  in  division  of  large  numbers  here  begun  should 
be  very  careful  and  methodical.  In  33  another  form  of  statement 
would  be  :  "  \  of  8  tens  of  sticks  is  how  many  ?  i  of  4  sticks  is 
how  many  ?  i  of  84  sticks  is  how  many  ?  "  This  process  is  some- 
times called  " partitio7i "  to  distinguish  it  from  "  division/'  or  the 
process  of  finding  how  many  times  one  number  is  contained  in 
another  numl)er. 

3G  Lead  the  pupils  to  use  tlie  objects  and  to  make  statements 
as  they  divide.  The  form  of  questions  used  for  exercises  on  page 
35  may  be  changed  to  asking  how  many  times  one  number  will  be 
contained  in  another  number.  For  example,  in  1 :  "  How  many 
times  are  2  sticks  contained  in  200  sticks  ?  How  many  times  are 
2  sticks  contained  in  20  sticks  ?  How  many  times  are  2  sticks 
contained  in  220  sticks  ?  " 

If  multiplication  has  been  taught  by  objects  thoroughly,  it  will 
not  be  necessary  to  continue  with  the  objects  in  division  to  a 
great  extent.  It  may  be  assumed,  for  example  in  6,  that  the 
pupils  know  that  2  is  contained  in  16  tens  8  tens  times,  and  all 
similar  examples  may  be  so  explained  without  tlie  aid  of  objects. 
In  8''  a  new  step  is  taken,  and  sliould  be  taught  with  objects  tlius  : 
"2  sticks  are  contained  in  13  tens  of  sticks  how  many  tens  times, 
and  how  many  tens  remainder  ?     2  sticks  are  contained  in  1  ten  or 

10  how  many  times  ?     What  is  the  answer  ?  "     This  principle  is 


III.  37]  TEACHEES     MANUAt.  43 

further  applied  in  ll,  and  a  similar  course  of  teaching'  should  be 
taken. 

After  the  examples  of  this  page  liave  been  performed  in  the  way 
indicated,  the  same  examples  may  be  performed  as  if  sticks  and 
not  times  were  called  for.  This  is  a  more  simple  process  to  teach. 
For  example,  in  1,  the  teacher  should  lead  the  pupils  to  get  ^-  of 
2  hundreds,  then  J  of  2  tens,  thus  finding  ^  of  220  to  be  1  hundred 
and  1  ten  or  110.  A  rapid  oral  review  without  objects  should  be 
given  before  the  next  page  is  taken. 

37  This  page  of  exercises  may  be  taken,  without  objects, 
orally.  In  such  exercises  as  23  a,  the  pupils  may  be  led  to  say 
first:  "4  in  120,  .30  times.  4  in  132,  33  times."  Afterwards 
they  may  perform  the  exercises  without  analysis. 

38  The  same  course  with  objects  is  to  be  pursued  here  as  was 
advised  for  the  exercises  on  page  36,  with  the  added  feature  of 
expressing  with  figures  the  results  as  they  are  found.  Teach  the 
terms  dividend,  divisor,  and  (piotlcnt,  and  lead  the  pupils  to  use 
them. 

39  Long  division,  if  thought  best,  may  be  begun  with  small 
numbers,  and  may  be  performed  without  objects.  A  good  form  of 
long  division  is  to  place  the  divisor  at  the  left  of  the  dividend,  and 
the  quotient  above  the  dividend,  as  shown  below. 

In  teaching  long  division,  the  i)upils  should  be  led  to  see  that 
there  are  no  new  processes  to  learn,  but   that  it  is  the  same  as 
short  division,  with  the  products  written  out  instead  of 
carried  in  the  mind.     The  following  order  of  teaching  is  ^^ 

suggested,  it  being  understood  that  as  the  pu])ils  answer        17)833 
the  questions  the  numbers  are  written.  68 

2  is  contained  in  4  tens  how  many  tens  times  ?     IT  is  153 

contained  in  83  tens  how  many  tens  times  ?     How  shall  153 

we  find  the  remainder  that  is  not  divided  by  17  ?    Multi- 
plying 17  by  4  tens  and  subtracting,  what  have  we  ?     This  number 
of  tens  and  3  units  is  Avhat  number  ?     17  is  contained  in  153  how 
many  times  ?     17  midtiplied  by  9  is  Avhat  ?     What  remainder  is 
there  ?     What  is  the  answer  ? 


44  GRADED    ARITH^NIETIC.  [ill.  40 

The  teaching  of  partition  in  "wliich  the  fractional  part  of  a 
number  is  found  is  also  to  be  taught,  although  the  form  of  Avork  is 
the  same.  The  statement  of  steps  taken  in  long  division  should 
be  at  first  very  simple,  following  somewhat  closely  the  order  of 
teaching  as  given  above. 


40    A7isu-ers:    1  22i|.     2  14 

e-     3  2H 

3-     4  20H- 

5  21i|. 

6  20i-o. 

7  20§f.       8  20§|. 

9  20§f. 

10  21^^,. 

11  21H. 

12  20§|. 

13  21§f     14  22if. 

15  21^V- 

16  22||. 

17  20i|. 

18  21|J. 

19  19|-|.     20  21if. 

21  23||. 

22  22||. 

23  16fi. 

24  16^^,. 

25  26J1-     26  14||. 

27  20/^. 

28  16i|. 

29  19|§. 

30  25|f. 

31  a  16H  ;    i^  25U ; 

c  43/^  ; 

d  50/5.       32  a  231 1  ; 

b  42/,  ; 

c  29/,  ;      cl  47if 

33  a  62/,: 

;       ^  49/^  ; 

c  50/5  ; 

d  2111. 

34  a  ll^V  ;    ^'  lOtI ; 

c  8||;     cZ 

lOH-       35 

76    24/,. 

36  50/^ 

52/,.       37  56}<i     31 

/,.       38  41/,     463A,. 

39  126. 

40  621. 

41  4(»|.        42  25|. 

43  18. 

44  15. 

45  73. 

46  49f. 

47  36i.       48  27|. 

49  25/0. 

50  27. 

Lead  the  pupils  to  divide  in  some  cases  by  a  trial  divisor,  con- 
sisting of  the  first  figure  of  the  divisor,  to  see  about  how  many 
times  the  divisor  is  contained  in  the  dividend.  This  will  be  more 
helpful  when  larger  numbers  are  used. 

41    A?iswers:   1  28|.        2  51i.        3  78|.        4  73*.  5  294. 

6  105.         7  115|.        8  123|.        9  llof  10  63if  11  68. 

12  112.      13  113f.      14  67/^.      15  134|.  16  442.  17  50/^. 
18  1351. 

Exercises  19  to  53  are  for  drill,  either  in  recitation  or  in  study. 
For  the  sake  of  accuracy  and  of  securing  practice  in  multiplication, 
it  is  well  to  have  their  Avork  in  these  exercises  proved  in  the  visual 
way,  —  multiplying  the  quotient  by  the  divisor  and  adding  the 
remainder  to  the  amount. 

43  Some  review  Avork  Avhich  the  pupils  slioiild  be  able  to  do 
without  lielp  of  any  kind.  Let  the  statement  of  steps  and  answers 
be  a  simple  and  natural  expression  of  the  pupils'  thought ;  thus,  in 
JL2  :    "  It  Avill  take  as  many  days  to  burn  35  cords  as  there  are 


III.  43]  teachers'    IMANUAL.  45 

sevens  in  35  "  ;  or,  '^  It  -svill  take  as  many  days  as  7  cords  is  con- 
tained in  35  cords "  ;  or,  ''  To  find  how  many  days,  I  divide  35 
cords  by  7  cords." 

43  Lead  the  pupils  into  the  habit  of  writing  out  the  steps  of  a 
problem  in  good  form,  and  also  of  marking  the  denomination  of 
each  number.  Two  forms  may  be  required,  • —  one  form  in  Avhich 
the  steps  are  only  indicated,  and  another  form  in  which  all  the 
steps  are  worked  out ;  thus,  in  1 : 

^^  of  396  mi.  =  average  number  of  mi.  in  1  da. 

18)  396  mi.  in  18  da. 
22  mi.  in    1  da. 

44  Extend  the  work  in  making  and  solving  original  problems. 
It  will  be  found  good  work  for  review. 

45  Some  teaching  of  tlie  writing  of  numbers  in  U.  S.  money 
should  precede  these  exercises.  Show  how  two  ounces,  eight 
pounds,  six  dollars,  may  be  expressed  by  figures  and  words,  and 
also  by  figures  and  abbreviations.  Give  the  sign  for  dollars,  and 
show  its  use  by  examples.  Lead  the  pupils  to  see  that  a  number 
of  cents  less  than  one  hundred  is  written  at  the  right  of  the  point, 
the  number  of  dimes  expressed  by  the  first  figure  at  the  right 
of  the  point,  and  the  number  of  cents  less  than  ten  expressed  by 
the  second  figure.  After  a  little  of  such  work,  the  pupils  ought 
to  be  ready  for  the  exercises  on  this  page. 

46  If  the  pupils  have  a  thorough  knowledge  of  writing  numbers 
in  U.  S.  money  they  should  have  little  difficulty  in  adding,  sub- 
tracting, multiplying,  and  dividing  such  numbers.  Before  the 
exercises  of  this  page  are  given  it  would  be  well  to  have  some 
simple  exercises  in  pointing  off,  such  as  the  following :  "  In  200 
cents  how  many  dollars  ?  Where  sliall  I  place  the  point  to  show 
the  number  of  dollars  ?  Point  off  the  dollars  in  300  cents  ;  220 
cents  ;  640  cents  ;  104  cents.  How  should  these  numbers  be  written 
for  addition  and  subtraction  ?  "  In  multiplication,  lead  the  pupils 
first  to  multiply  dollars,  then  cents,  and  finally  dollars  and  cents, 
such  as  the  following  :    $84  X  4  ;    |61  X  8  ;  $75  X  6  ;  $0.80  X  2  ; 


46  GRADED    ARITHMETIC.  [III.  47 

$0.75  X  4  ;  $1.50  X  6  ;  $2.24  X  3  ;  $1.08  X  4  ;  $1.38  X  6.  In 
this  work,  insist  upon  the  point  being  made  in  all  cases,  and.  lead 
the  children  to  see  that,  if  the  multiplicand  is  cents  the  product 
will  be  cents,  and  that  it  must  be  pointed  off  accordingly. 

It  is  well  in  the  written  statement  of  the  solution  of  problems  to 
write  what  is  given  and  what  is  required  to  be  found ;  thus,  in  1 : 

Given  the  cost  of  1  cow  and  1  horse. 
To  find  the  cost  of  6  cows  and  3  horses. 

$64  -cost  of  1  cow.  $140  cost  of  1  horse. 

6  3 

$384     ''     "  6  cows.  $420     "     "  3  horses. 

420     "     "  3  horses. 
$804     "     ''  6  cows  and  3  horses. 

47  Before  giving  5,  6,  7,  and  8  to  the  pupils,  lead  them  to  see 
that  in  all  such  problems  the  dividend  and  divisor  must  be  of  the 
same  denomination.  This  may  be  done  by  showing  that  2  cents 
is  contained  in  6  dimes  or  6  dollars  more  than  3  times.  Give  a 
few  exercises  like  the  following :  "  3  cents  in  3  dimes  how  many 
times  ?  6  cents  in  3  dimes  ?  2  dimes  in  1  dollar  ?  4  dimes  in 
2  dollars  ?  5  cents  in  2  dollars  ?  4  cents  in  1  dollar  ?  4  cents 
in  8  dollars  ?  "  9  and  10  should  be  performed  and  explained  by 
partitioTi,  in  which  the  answer  is  the  same  denomination  as  the 
sum  divided;  thus:  i  of  $4  =  $2;  I  of  $8.25  =  $1.65.  In  these 
and  similar  exercises  pay  particular  attention  to  pointing  off.  In 
exercises  like  the  last  of  10  it  may  be  Avell  to  reduce  the  sum 
divided  to  cents  before  dividing  ;  thus:  "  jL  ^^  '^^^  cents  is  34  cents." 

48  Let  the  pupils  perform  as  many  of  these  orally  as  they  can, 
1,  2,  and  4  to  be  done  by  partition,  and  3  by  division.  A  simple 
statement  of  process  should  be  made  to  represent  the  difference 
between  these  two  operations  ;  thus,  in  "L  <( :  "1  qt.  of  kerosene 
will  cost  i  as  much  as  4  qt.  ^  of  36/  =  9/."  And  3:  "Since  1 
"book  can  be  bought  for  25/,  as  many  books  can  be  bought  for 
$4.25  as  thei'e  are  twenty-fives  in  425,  or  17  books";  or,  '<as  many 
books  as  25/  is  contained  times  in  425/." 


III.  49]  I'EACHERS'    MANUAL.  47 

If  the  use  of  fractions,  such  as  are  given  in  6  and  7,  is  not 
familiar  to  the  pupils,  give  some  simple  work  with  blocks,  as  ^ 
of  16  blocks,  f  of  IG  blocks,  |-  of  20  blocks.  Discs  or  drawings 
of  circles,  squares,  or  lines  may  be  used  in  finding  the  fractional 
part  of  numbers  that  cannot  be  evenly  found  ;  thus  :  ^  of  4  circles, 

1  of  4  circles,  ^  of  6  circles,  f  of  6  circles,  etc.,  may  \w  taught. 
49-51    No  new  principle  is  involved  in  these  exercises.     State- 
ments like  the  following  may  be  placed  sometimes  before  problems, 
so  as  to  make  their  conditions  more  clear  to  the  pupils,  thus  : 

.    36  oranges  cost    72/  1  yd.  costs    14/ 

1  orange    costs  ?  ?    "    cost    224/ 

It  is  well  occasionally  to  ask  such  questions  as:  "8  lb.  will  cost 
how  many  times  as  much  as  1  lb.  ?    how  many  times  as  much  as 

2  lb.  ?  how  many  times  as  much  as  4  lb.  ?  WJiat  part  as  much 
as  16  lb.":'"'  Pupils  should  be  able  to  recognize  from  the  first  that 
a  fractional  part  of  the  given  number  is  taken  when  the  required 
answer  is  of  the  same  denomination  as  tlie  number  divided,  and 
that  division  of  one  number  by  another  is  performed  when  the 
number  of  parts  is  required. 

53  Great  care  should  be  taken  in  making  these  bills,  directions 
being  given  as  to  ruling,  date,  receipt,  etc.  The  pupils  should 
learn  to  make  their  own  rulings. 

53  Toy  money  may  be  first  iised  in  performing  these  exercises, 
if  necessary.  Pupils  should  be  able  to  tell  instantly  the  difference 
between  one  dollar  and  any  sum  of  money  below  that  amount,  thus 
avoiding  the  necessity  of  adding  in  making  change. 

54  As  many  of  the  problems  in  this  section  as  possible  should 
be  performed  orally.  Some  of  them  may  be  written  out  for  the 
sake  of  practice  in  written  analysis.  The  making  of  original  j'rob- 
lems  may  be  considerably  extended. 

55-64  Weights  and  measures  should  be  used  in  performing 
the  problems  of  this  section  when  needed.  It  Avill  be  necessary 
to  use  them  considerably  if  they  have  not  been  used  before  ;  and 
if  they  have  been  used  before,  the  pupils  should  be  led  to  Avork 


48 


GRADED   AEITHMETIC. 


[III.  58 


■with  the  measures  or  drawings  of  the  measures  whenever  the  con- 
ditions of  a  problem  are  not  clearly  understood.  Nearly  all  of 
these  problems  should  be  performed  without  figures ;  but  for  prac- 
tice in  stating  the  steps  of  the  solution,  some  of  them  may  be 
written  out  in  full.  It  is  advisable  for  the  teachers  to  spend  some 
time  in  showing  the  pupils  a  good  form  of  solution.  The  following 
written  solutions  are  suggested  : 

Exercise  10,  page  58. 

Given  the  number  of  bushels  of  grain  on  hand. 

To  find  the  number  of  bushels  more  to  fill  an  order. 

1000  bu.  of  grain  ordered. 
735  "  "  ''  on  hand. 
2r7     '^     "       ''     required  to  fill  the  order. 


Exercise  6,  page  62. 

Given  the  amount  of  paper  a  bookseller  had  and  sold. 
To  find  the  amount  he  had  left. 

10  reams  12  quires  =  212  quires,  amt.  of  paper  on  hand. 
8      "        4      "      =164       "  ''      ''       "      sold. 


48 


remaining. 


If  thought  best,  the  parts  of  the  solution  may  be  ruled  off  in  the 

Exercise  13,  page  63. 


following  manner 


Required. 

Solution. 

Answer. 

30  lb.  in  1  tub. 

28   ''    «'  1     '' 

58   ''    ''  2  tubs. 

Cost  of 

$0.13  cost  of  1  lb. 

$7.54 

Sugar. 

X58 
104 
65 

$7.54  cost  of  58  lb. 

IIT.  65]  teachers'  manual.  49 

G5-07  The  pupils  have  had  some  practice  in  measuring,  but 
it  should  not  be  presumed  that  they  do  not  need  more  practice  of 
the  same  kind.  In  addition  to  tlie  writing  out  of  steps  in  the 
solution  of  problems,  the  pupils  should  be  led  sometimes  to  draw 
a  diagram  representing  the  distances  given  and  required.  This 
should  be  done  in  such  problems  as  4,  6,  8,  10,  11,  12,  and  16, 
page  67. 

6^^—70  Do  not  permit  pupils  at  this  stage  to  find  the  square 
contents  of  a  rectangle  by  multii)lying  the  length  by  the  width. 
Lead  them  always  to  think  of  the  number  of  units  in  a  row  to 
be  multiplied  by  the  number  of  rows.  In  the  exercises  on  page  69 
the  required  measurements  should  be  made,  but  tlie  fraction  of 
an  inch  or  foot  may  be  omitted.  Wherever  it  can  be  done,  a  plan 
of  the  surface  should  be  drawn  to  scale.  This  should  be  done  in 
nearly  all  of  the  problems  on  page  70.  Oral  and  written  state- 
ments of  processes  should  be  required.  For  example,  in  7,  page  70, 
the  oral  statement  might  be  :  "  In  the  passage-way  there  are  10 
rows  of  blocks  and  12  blocks  in  a  row.     Since  in  1  row  there  are 

12  blocks,  in  10  rows  there  are  10  times  12  blocks,  or  120  blocks." 
71-73    If  numbers  to  lOOO,  including  the   four  fundamental 

rules,  have  been  thoroughly  taught  by  objects,  the  knowledge  thus 
gained  will  serve  as  a  foundation  for  the  teaching  of  numbers  in 
the  higher  orders.  At  this  point  the  pupils  are  supposed  to  know 
that  ten  of  any  order  make  one  of  the  next  higher  order,  and  that 
the  value  of  the  number  expressed  is  determined  by  the  position 
that  the  figure  has  with  reference  to  the  decimal  point.  Proceeding 
upon  this  basis,  the  pupils  count  the  thousands  precisely  as  the 
units  were  counted,  until  a  thousand  thousand  is  reached,  when 
the  number  is  called  one  million.  The  period  of  thousands,  both 
in  reading  and  writing,  should  be  treated  exactly  as  the  period  of 
units  was  treated,  ten  of  each  order  making  one  of  the  next  higher. 
Extend  the  work  here  given  if  necessary. 

74    Answers:   9  7703.       10  15044.       11  17788.       12  11125. 

13  13888.  14  18054.  15  20089.  16  20958.  17  16987. 
18  13587.       19  25294.       20  11896. 


75    Ansiv 

ers  : 

1  104 

5 

8003. 

6    1 

5750. 

10 

110981. 

11 

51720. 

15 

765277. 

16 

07688 

20 

10042. 

21 

9558. 

50  GRADED   ARITHMETIC.  [III.  75 

The  combinations  in  thousands  will  be  found  easy,  if  the  corre- 
sponding combinations  in  the  period  of  units  are  well  understood. 
The  first  eight  exercises  can  doubtless  be  performed  mentally  by 
the  pupils,  but  for  practice  in  expression  the  numbers  sliould  be 
written  in  full.  For  convenience  the  fifth  and  sixth  orders  should 
be  named  ten-thousand  and  hundred-thoasand.  Tlie  reason  for 
carrying  one  for  every  ten  of  a  loAver  order  should  be  given  in  the 
same  way  as  was  taught  for  numbers  below  1000.  For  example,  in 
15,  the  pupil  is  led  to  say:  "8  and  3  and  9  hundreds  is  20  hun- 
dreds, equal  to  2  thousands,  to  be  added  with  the  thousands.  2,  8, 
12,  20  thousands,  expressed  as  2  ten-thousand,  and  0  thousand,"  etc. 

I.        2  15749.        3  19378.        4  14574. 

7    233269.         8    93533.         9    54877. 

12  674673.      13  73,3299.      14  884770. 

17  68185.       18  155813.       19  15403. 

22  10688.  * 

Let  tlie  statement  of  steps  in  adding  be  continued  until  the 
pupils  have  a  clear  idea  of  the  process.  Give  frequent  exercises 
in  dictating  numbers  for  addition.  This  is  best  for  seat-work,  so 
as  to  be  sure  that  all  work  independently. 

76    Ansivers:    1  218495.       2  11084.       3  a  12734;     h  16404 
c  21729  ;     d  15131 ;     e  25809  ;    /  20045  ;     y  22148  :      h  25484 
i  35362  ;    J  14794  ;     k  17256 ;     I  18801 ;     m,  22300.       4  96513 
5  549.        6  1378.        7  2321.        8  1047.        9  7742.        10  6942 
11  2290      3144      1480      1930.        12  3250      2173      3760      3236 
13  2417     751     8690     86     6048     9099     3215.        14  2468     3239 
8189      5686      7005      9012      1021.        15  267      820      8176      575 
7058     4639     4423.        16   8392      6401      8355     672     25(5     4165 
312.        17   3685     517     254     833     1158     48.         18   3046     5995 
1045      8401      6355      715.        19    4915      3289      786      8387      704 
1653.       20  1007     5127     7564     69     859     6444. 

Lead  the  pupils  to  give  a  statement  of  steps  in  the  process  of 
subtraction  until  it  is  ck^arly  understood.  If  the  process  is  not 
understood,  use  objects  in  the  subtraction  of  numbers  to  1000. 


III.  77]  teachers'  manual.  51 

77  Ans7vers:   1  1G82.       2  5343.       3  652.       4  642.       5  604. 

6  3264.  7  5377.  8  7077.  9  6398.  10  235.  11  3237. 
12  2353.  13  2318.  14  2161.  15  omn.  16  rAuA).  17  294. 
18  5659.  19  969.  20  8449.  21  11456.  22  12122. 
23  26808.  24  28539.  25  23999.  26  6588.  27  23175. 
28  3179.  29  181044.  30  273706.  31  93341.  32  588680. 
33  175629.       34  302357.       35  301245.       36  176586. 

Let  the  pupils  prove  their  answers.  Show,  by  use  of  small 
numbers,  why  the  sum  of  the  subtrahend  and  remainder  ought  to 
be  the  same  as  the  minuend. 

78  Ansivers:  1  a  633;  h  203;  c  25  ;  d  1225;  e  192; 
/  873  ;  (/  6566  ;  7i  92  ;  i  5275  ;  j  2569  ;  k  666.  2  a  138  ; 
b  3002  ;  c  25;  (I  244  ;  e  733  ;  /  2587  ;  f/  1316  ;  h  91  ; 
i  929;  ./  609;  A-  177.  3  a  2445;  ^-  252 ;  c  6342;  d  307; 
e  86  ;  /  579  ;  rj  U  ;  h  552  ;  i  2239  ;  j  1326  ;  /.:  212. 
4  81208.  5  25156.  6  7992.  7  8925.  8  77611.  9  92316. 
10  11102.       11  6966.       12  17212.       13  10628. 

The  proof  by  subtraction  is  only  given  for  practice  in  subtrac- 
tion. The  best  proof  in  addition  is  made  by  adding  columns  down 
if  they  have  been  added  up. 

79  Ansicers:  1  a  25483;  h  7233;  c  20319;  d  22857 
e  7866;  /  38134;  r/  21684  ;  A  14280  ;  i  8098  ;  y  1582 
2  /  4074  ;  g  7123  ;  h  5003  ;  i  2110  ;  _/  20.  3  /"  2229 
y  5175;  h  3922;  i  1618;  ,/  122.  4  /  1774;  r/  435 
h  1046  ;  I  9!)4  ;  J  381.  5  /  5185  ;  y  5269  ;  h  2287 
i  2249  ;   ,/  1.         6  ('  2066  ;    b  5115  ;    c  2169  ;    d  3608  ;    e  3592 

7  "  2509  ;  h  289  ;  c  1542  ;  d  3023  ;  e  41.  8  «  2912 
I>  119;  c  2423  ;  d  383  ;  e  345.  9  "  2282  ;  ^.  192  ;  r  1688 
(Z2301;    e53.     10  2918  mi.     1152768.     12  30961.    13  164932. 

80  After  these  exercises  have  been  performed  orally,  it  may 
be  well  to  express  the  work  of  some  of  them  in  figures.  Such  exer' 
cise  will  prepare  the  pupils  for  subsequent  work, 


52 


GRADED    ARITHMETIC. 


[III.  81 


81    Answers:    1  3474.  2  12274.  3  20032.  4  102848. 

5  816458.          6  28452.  7  81492.  8  94542.  9  112208. 

10  753354.       11  37536.  12  82768.  13  102816.  14  57312. 

15  42840.  16  45474.  17  27870.  18  127224.  19  39702. 

20  44520.  21  11572.  22  18972.  23  17955.  24  17088. 

25  14382.  26  24783.  27  26828.  28  36868.  29  11025. 

30  25826.  31  48802.  32  28006.  33  34941.  34  33579. 

35  55476.  36  21924.  37  66720.  38  46315.  39  38016. 

40  66690.  41  49068.  42  23374.  43  76665.  44  68992. 

Before  multiplying  by  tens  and  units  as  given  in  the  exercises 
beginning  Avitli  12,  let  the  pupils  have  some  review  in  multiplying 
smaller  numbers  with  objects,  to  show  that  the  right-hand  figure  in 
multiplying  by  tens  is  in  the  tens'  column. 


83    Answers 
5    19300.  ( 

10  $1008.25. 
14    $2.88. 
18    $4148.16 
22  $137.19. 
$8131.02     $7503.75 
h  $2135.60    $3155.90 
$2484.44    $2745.50 


I  4760.         2   2140.         3   30000.  4   86850. 
5    29940.          7    5160.          8    80480.  9    32100. 

II  $4191.50.         12    $85.68.         13  $1402.56. 
15    $9011.25.         16    $1083.06.         17  $4567.50. 


19    $5984.94.  20    $506.  21   $92.64. 

23  $48.90.      24  a  $4889.40     $7225.35     $6575.46 

$6987.84     $5688.06 

$3551.48 

c  $5295.20 


$2872.04 
$2661.14; 
$7710.72 

$3401.10 
$3151.35  ; 

$5542.08 


$6276.48 


>972.16  $8280 

d  $2529  $3737.25 

$2942.10  $3251.25 

$6448.74  $5951.25 
/  $4158.80    $6145.70    $5592.92 

$4838.12  $5346.50    $5182.22; 

$5514.14  $5088.75     $4738.88 

25  $228.  26  $10560.       27  $231904. 


;6285.75     $6092.61  ; 

^3277.50     $3052.16 

$7972.80    $7255.68 

$6836      $6722.88 ; 

$4205.70      $3881.25     $3614.40 

e  $3867.80    $5730.45    $5215.02 

$4511.22     $4985.25    $4832.07; 

$6916.04    $6382.50     $5943.68 

g  $3305.80    $4899.95    $4459.22 

$3857.42     $4252.75     $4131.77. 


83  Some  of  these  exercises  should  be  performed  by  the  aid  of 
figures,  and  each  step  explained,  as  for  example,  14,  in  which  the 
pupihs  maybe  led  to  say  :  "16  nnits  in  48  units,  3  times  ;  16  units 
in  48  tenSj  3  tens  or  30  times  ;  16  units  in  48  hundreds,  3  hundreds 


III.  84]  teachers'  manual.  53 

or  300  times  ;  16  units  in  48  thousands,  3  thousands  or  3000  times." 
Some  of  the  exercises  should  be  performed  by  partition  as  in  1 : 
^  of  40,  ^  of  400,  etc. 

84: .  A?i.mwrs :  1  S71.  2  1414.  3  1593.  4  2347.  5  1982. 
6  70().  7  1299.  8  3276.  9  14521.  10  176(50.  11  4581. 
12  <'.7()9.  13  2509 J.  14  3466.  15  1226.  16  13801. 
17  12562.  18  316f  19  8201.  20  9102.  21  1243. 
22  12439.  23  5S00.  24  .S098.  25  276.  26  568.  27  342. 
28  172.  29  112.  30  258.  31  763.  32  54.  33  63. 
34  40||.  35  514-^:j.  36  1058^9^..  37  112||.  38  Si'l/^-. 
39  474/g.  40  4652j\.  41  15742^.  42  79O5!'..  43  5J. 
44  88.  45  61.  46  25.  47  201f.  48  1151.  49  84§-3. 
50  39] 711.  51  1540i§.  52  1234ff  53  493?i.  54  799a^. 
55  693fi.  56  163.  57  104^^.  58  1138JL.  59  923§|. 
60  71735^.  61  1264if.  62  1055/g.  63  27.  64  2306. 
65  48.       66  S%\%.       67  715|f.       68  627J5. 

85  Answers:  1  10070.  2  6410ffi.  3  168|i.  4  3027^- 
5  3280||.        6  5768i§.        7  148|^.        8  1227||.        9  19140;fV 

10  1030.  11  2004.  12  470.  13  3005.  14  1608.  15  1020. 
16  26.  17  368.  18  275.  19  26.  20  lOOOjf.  2I  13030. 
22  38886.  23  28684.  24  164356;  25  21834.  26  14692. 
27  9068.  28  6912.  29  1719.  30  2440.  31  3280. 
32  10495.       33  11076.       34  57631.       35  17526. 

If  the  finding  of  fractional  parts  of  numbers  is  not  understood, 
teach  the  process  Avith  objects,  using  small  numbers. 

80  Answers:  1  958.  2  1480.  3  121522^^.  4  1093^V 
5  32713.        6  3472.        7  10200.        8  68.        9  125.       10  fO.78. 

11  1^13.50.  12  $493^.  13  $125.37.  14  $45.75.  15  $35.27. 
16  105  yds.  17  1376  acres.  18  837  pairs,  $0.70  left. 
19    897  quires. 

Call  attention  to  the  fact  that  to  have  an  entire  number  of  parts 
in  the  quotient,  the  dividend  and  divisor  must  be  of  the  same 
denomination  ;  and  that  in  getting  the  fractional  part  of  a  number 


54  GRADED    ARITHMETIC.  [III.  87 

the  answer  is  of  the  same  denomination  as  the  number  wrought 
upon.  Problems  like  10  to  15  may  be  performed  by  taking  the 
fractional  part,  thus : 

$74.10  =  cost  of  95  bu. 


1 

^5 


of  $74.10=    "     ''     1  hn.  =  ^'l     =$0.78. 

9o 


87  Answers:    6    $3.25      $40,625.         7    20   kegs.       8    70    T. 

9  6000  pkg.       10  $221.25.       11  $6.75.       12  13  T.  14  lb.  10  oz. 
13  17  T.  9  lb.  11  oz. 

Encourage  originality  in  the  solution  of  problems  like  6.  Some 
pupils  may  find  the  cost  of  1  cwt.,  and  then  of  5  cwt.  Others  may 
regard  the  5  cwt.  as  ^  of  a  ton  and  multiply  by  6^.  Lead  them 
to  take  the  shorter  way  whenever  it  is  clearly  understood. 

Lead  the  pupils  to  see  that  the  same  principle  is  involved  in  the 
addition  of  compound  numbers  as  in  the  addition  of  simple  num- 
bers, the  only  difference  being  in  the  system  of  notation. 

88  Ansivers :  1  10  lb.  2  7  lb.  1  oz.  3  10  lb.  3  oz.  4  8  T. 
5  10  T.  100  lb.       6  15  T.       7  1058v§.       8  996  mi.       9  293662. 

10  $4166.66|.         11  357866.         12  2286  ft.       634  ft.       365  ft. 
11383  ft.  combined  height.       13  3280  ft.       14  177  ft.  less. 

89  Ansivers:  1  $2724.  2  $3364.28.  3  $95.19.  4  $595.35. 
5  $46494.  6  $11319.  7  5401  gal.  8  2894  ft.  9  6  yr. 
10  152  mi.       11  $1552.96.       12  $234. 

90  Answers:  1  $586.2.58.  2  $1750.  3  84  casks.  4  357/^ 
:bars.     5  245  yds.     6  2297.     7  $3841.25.     8  48  tubs. 

Some  of  the  most  difficult  of  the  problems  on  the  last  pages  of 
this  section  may  need  to  be  talked  about  in  tlie  recitation  before 
they  are  given  as  a  lesson  for  the  pu])ils  to  learn.  Such  questions 
as  the  following  for  2,  page  90,  may  be  helpful  in  inducing  the 
pupils  to  think,  and  in  leading  them  to  give  a  good  written  analysis: 
"What  is  given  in  this  problem?  What  is  required?  Do  you 
know  the  whole  number  of  through  passengers  ?     How  can  you 


III.  91]  teachers'  manual.  56 

find  the  number  ?  How  can  you  find  the  whole  amount  paid  by 
these  passengers  ?  " 

91—101  Whenever  the  conditions  of  a  problem  are  not  clearly 
understood,  lead  the  pupils  to  grasp  them  by  the  use  of  questions 
and  illustrations,  and  not  by  any  set  form  of  reasoning.  Let 
the  questions  be  such  as  to  make  the  pupils  think.  In  such  prob- 
lems as  7,  page  91,  some  preliminary  questions  like  the  following 
may  be  helpful :  "  10  pounds  cost  what  part  as  much  as  20  pounds  ? 
4  pounds  cost  what  part  as  much  as  20  pounds  ?  Do  you  see  now 
any  way  of  getting  the  cost  of  8  pounds '/  "  If  the  pupil  is  still 
uncertain,  ask  him  what  4  pounds  cost,  and  then  what  8  pounds 
cost.  In  finding  the  cost  of  25  pounds,  the  pupils  may  be  led  to 
find  the  cost  of  1  pound  first,  and  then  of  25  pounds.  There  is 
some  advantage  in  having  pu})ils  form  a  habit  of  working  through 
the  unit  in  finding  the  cost  of  a  given  number.  But  in  such  prob- 
lems as  6,  page  91,  it  is  better  to  work  by  multiples.  To  perform 
this  problem  the  pupils  should  be  led  to  see  that  12  peaches  will 
cost  6  times  as  much  as  2  peaches. 

In  some  problems  it  may  be  well,  if  the  pupils  find  difficulty, 
to  lead  up  to  the  required  result  by  carefully-graded  steps  ;  for 
example,  in  3,  page  92,  to  ask  how  many  eggs  Avould  pay  for  40 
cents'  worth  of  butter,  80  cents'  worth,  etc.  In  1,  page  95,  the 
questions  might  be  :  "  How  many  can  I  make  in  1  hour  ?  in  3 
hours  ?  "  And  in  5  :  "  How  many  times  can  the  measure  be  filled 
from  a  quart  can  ?  from  a  gallon  can  ?  from  a  two-gallon  can  ?  " 
In  such  problems  as  4,  page  98,  and  2,  page  99,  it  will  be  found 
useful  to  give  the  same  conditions  with  small  numbers.  Formal 
oral  '<  explanations "  of  problems  should  not  be  required  at  this 
time.  Statements  of  processes,  however,  may  be  made,  but  care 
should  be  taken  that  the  words  exactly  represent  the  thought  of 
the  speaker  with  little  reference  to  the  form  of  language. 

Continued  attention  should  be  paid  to  the  written  analysis  in 
the  solution  of  problems.  The  following  analyses  of  the  last  three 
problems  on  page  101  may  suggest  good  forms  for  the  pupils : 


56  GRADED   ARITHMETIC.  [III.  101 

Given  the  number  of  bu.  in  4  bins. 
To  find  the  number  of  lb.  in  4  bins. 

"5  bu.  60  lb.  in  1  bu. 

48    "  _248 

90    ''  480 

35    '■'  240 


248    '^    in  4  bins.  120 


14880  lb.  in  248  bu. 

Given  the  height  of  an  iceberg  above  water. 
To  find  the  Avhole  height  of  the  iceberg. 

612  in.  =  height  of  iceberg  above  water. 
8 


4896  in.  =       "       •<        "       under      " 

612  in. 
5508  in.  =  entire  height  of  iceberg. 

Given  the  cost  of  1  ton  of  coal. 
To  find  the  cost  of  87  tons. 

$7.25  cost  of  1  T. 

87 
5075 
5800 


$630.75  cost  of  87  T. 


SECTION   VI. 

NOTES   FOR    BOOK   NUMBER   FOUR. 

A  mastery  of  the  subjects  presented  in  Book  No.  4  will  give 
nearly  all  the  practical  knowledge  of  Arithmetic  needed  for  the 
ordinary  affairs  of  life,  besides  furnishing  a  good  foundation  for 
subsequent  work.  It  may  not  be  necessary  for  pupils  to  perform 
all  the  examples  and  problems  here  given  ;  but  before  any  consider- 
i;bh  r'lir.ber  of  them  are  omitted,  the  teacher  should  be  sure  that 


IV.  1]  teachers'  manual.  57 

they  are  not  needed,  either  for  the  purpose  of  fixing  the  principles 
and  processes  which  have  been  taught,  or  for  the  purpose  of  mental 
discipline. 

To  more  clearly  understand  the  right  use  of  the  book,  teachers 
are  advised  to  read  the  Note  to  Teachers,  in  which  are  given  its 
distinctive  features  and  some  hints  as  to  possible  dangers. 

Appliances.  —  Some  of  tlie  appliances  needed  for  teaching  the 
various  subjects  may  be  supplied  by  the  teacher  and  pupils  as 
they  are  needed,  but  it  would  be  well  to  have  at  the  outset  all 
the  common  weights  and  measures,  and  plenty  of  cardboard  or 
old  pasteboard  boxes  from  which  discs  and  squares  may  be  made. 

Analysis  of  Problems.  —  Ko  formal  explanations  or  reasons  should 
be  insisted  upon,  but  tlie  method  of  solution  should  be  frequently 
called  for,  both  of  oral  and  of  written  problems.  The  aim  should 
be  first,  to  have  the  pupils  think  as  they  solve  the  problem,  and 
secondly,  to  have  them  express  their  thoughts  in  their  own  words. 
Good  written  forms  of  analysis  should  be  required. 

Development  "Work.  —  It  has  been  the  aim  to  give  many  simple 
exercises  leading  up  to  difficult  processes  or  jDrinciples.  If  any 
problem  is  found  too  difiicult  for  the  pupils  to  perform,  instead 
of  attempting  to  "explain"  the  problem  in  words,  teach  the  part 
not  understood  by  the  use  of  illustrations,  or  lead  up  to  it  by  simple 
questions  involving  small  numbers. 

1—7  A  few  of  these  problems  may  be  found  too  difficult  to  be 
performed  orally,  but  let  the  pupils  try  them  in  that  way  before 
the  pencil  is  taken.  Not  much  development  work  ought  to  be 
needed  for  these  review  problems.  Possibly  in  such  problems  as 
7,  page  5,  some  questions  like  the  following  may  be  asked  ;  "  If 
there  were  3  rows  of  trees  and  4  trees  in  a  row,  and  the  trees 
were  placed  20  ft.  apart,  how  long  and  wide  would  the  lot  be  in 
which  the  trees  are  planted  ?  How  many  feet  of  fencing  would 
be  needed  for  such  a  lot  ?  Suppose  there  were  8  rows  and  6  trees 
in  a  row,  how  long  and  wide  would  the  lot  be  ?  How  many  feet 
of  fencing  would  be  needed?"  Such  questioning  may  not  be 
needed,  especially  if  the  pupils  are  required  to  draw  an  illustrative 


58  GRADED    ARITHMETIC.  [IV.  8 

diagram.  5,  page  6,  is  of  a  different  kind,  and  may  need  such 
questions  as  :  "  Hoav  many  miles  in  1  hour  ?  in  3  hours  ?  in  8 
hours  ?  How  many  hours  would  it  take  him  to  walk  3  miles  ? 
6  miles  ?  12  miles  ?  "  Simple,  natural  statements  of  processes 
should  be  expected,  but  see  that  they  are  not  too  wordy  and 
labored. 

8  A7iswers:  1  1303.31.  2  $643.38.  3  $899.20.  4  $1235.80. 
5  $1192.34.     6  $6303.68.     7  $3293.83.     8  $5681.13.     9  44214. 

10  174596.  11  $2533.83.  12  $2266.11.  13  $1565.12. 
14  $1892.34. 

9  Ansu-ers :  1  $59.40.  2  $317.  3  $535.70  a  mo.  $20.6038 
a  day.  4  $657.  5  $336.  6  $57.80.  7  $195.50.  8  $5.18. 
9  $9.52.  10  $0.24.  11  $2.  12  $60.75.  13  $54.40. 
14  9.46  bbl.       15  $115.39.       16  2920  lb.     Ifa  T. 

10  Aiiswers:    1  25110  sq.  ft.         2  180  ft.         3  1422f  sq.  yd. 

4  $697.  5  $5.55.  6  $66.24.  7  $33.60  gain.  8  $1561.91. 
9  $981.34.       10  $1080.11.       11  $1977.29. 

11  Answers:    1    8580.        2    2630.        3    10520.        4    42900. 

5  44125.  6  5150.      7  30900.      8  28504.      9  3483^.      10  3914. 

11  1265.  12  368.  13  6  yr.  10  mo.  14  243.  15  $0.35. 
16  96  bu.  17  $64.50.  18  576  lb.  19  $66.64.  20  $7390.50. 
21  $4.44. 

13  Answers:  1  $123.70.  2  17600yd.  3  $54.40.  4  150 
times     960  times.       5  5000  min.       6  55|  yd. 

In  all  problems  on  the  last  five  pages  of  this  section,  require 
the  pupils  to  write  out  statements  of  processes,  to  draw  diagrams 
when  needed,  and  to  carefully  label  each  result. 

13  A  brief  exercise  from  the  board  to  teach  orders  and  periods 
may  be  given  before  the  pupils  answer  the  questions  given  on  this 
page.  The  exercises  on  this  page  will  suggest  the  kind  of  work 
to  be  given. 

14  Much  drill  similar  to  this  should  be  given  from  the  board 
and  by  dictation.     Observe  the  omission   of  the   word   ''and"  in 


IV.  15] 


teachers'  manual. 


59 


reading  integers.     Constant  care  in  this  regard  will  prevent  con- 
fusion in  the  reading  of  mixed  decimals. 


1    4437920. 
5    7261543. 


2    44201673. 
6    5051010. 


3 
7 


3348952. 
1201991. 


15    Ansivers 
4    11888983. 
8   990990991. 

If  the  pupils  have  been  thoroughly  drilled  in  writing  numbers, 
and  if  they  know  well  the  orders  and  periods,  there  will  be  little 
difficulty  in  performing  these  exercises. 


16    Ansivers:     1    79615294. 


4    71053996. 
8  44839931. 
12    114480. 
16    5005625. 
20    359406. 
24    674700. 


5  1500635. 
9   357000879. 

13  183736. 
17    2385184. 

21  306000. 
25    218772. 


2    2299711. 
6    174679994. 
10   56490661. 
14    430008. 
18    255816. 
22    154830. 
26    350520. 


3    5392649. 

7    17391047. 

11  271368. 

15    2609088. 

19    413100. 

23     465885. 


If  pupils  thoroughly  understand  and  can  explain  the  four  funda- 
mental processes  to  millions,  no  explanation  of  those  processes  in 
the  higher  orders  need  be  required. 


17    Answers:      1     9746388         18174424 


9  184. 

14 

19 


2  293900640 

3  9940860 

4  217914192 

5  230709840 

6  574038864 

7  585462987 

8  155670762 
10  127^V 


1287i| 


24 
30 
35 
39 


15  854|i 
20 
200000.  25 
1045.  31 


9445f| 


6334|3. 
2000. 


-|QA714  0 


408789072 
11357820 
274410464 
1141472235 
558977328 
553881996 
112108854 

11  174-1*. 
16  879ig. 

21  6835f§. 


21658640. 
608285264. 
22169520. 
6505142176. 
3416445000. 
1620167808. 
797622465. 
351392574. 
12  130§f.    13  93af 
17  645^.   18  862/^ 
22  4296i6j- 


23  96332f. 


26  200 
32 


3010fg 


6392UI. 


36 


1922  9  0 


40 


1867311^. 


27  3000 
33 
37 

41  15014-11  f 


851/A 
8o2§tf 


28  300. 
34 


29  801. 

4148^-^0- 
38  877^3^^. 
42  10290^^^. 


60  GRADED    ARITKMETIC.  [IV.  18 

43  2o742§f 0.         44  7121i|j.  45  SSeifff.  46  333f§^f. 

47  442H§4.           48  431tVtV-           49  410^^2^-  50  92f|3. 

51  483i§|.  52  830f34o.  53  SJO^U-  54  UOS^b^. 

55  901fej.  56  401§ff|.  57  2423iif|.  58  762f ^sa. 

18  Ansivers:  1  11514500  sq.  mi.  2  1940162  3571392 
2994002.  3  774  mi.  1278  mi.  525  mi.  4  35.2  mi.  Omaha, 
12  h.  26  m.  P.M.  Tuesday  ;  Cheyenne,  3  h.  38  m.  a.m.  Wednesday ; 
Ogden,  8  h.  38  m.  p.m.  Wednesday;  Palisade,  5  h.  17  m.  a.m.  Thurs- 
day; Reno,  1  h.  16  m.  p.m.  Thursday;  Sacramento,  5  h.  38  m.  p.m. 
Thursday;  San  Francisco,  8  h.  11  m.  p.m.  Thursday. 

Other  problems  similar  to  these  and  to  the  problems  on  the 
following  pages  should  be  given,  the  object  being  to  combine  the 
getting  of  useful  information  with  work  in  Arithmetic.  The  in- 
formation may  be  gathered  from  the  regular  text-books,  almanacs, 
newspapers,  and  railroad  guides. 

19  Answers:  1  $162065.  2  $121481.  4  $1545777816. 
5  $2708393886  $567934781  $84886022  $236244642  $53642805 
$129756975     $145141038     $54989193     $1162616070. 

For  areas  referred  to  in  3,  see  tables  given  on  pages  20  and  21. 
30 


.•  1  Area,  162038 

Population, 

17401545. 

2      268620 

8857921. 

3      610215 

"  11150675. 

4      753540 

22362279. 

5     1601103 

3059408. 

6     3395516 

62831828. 

7  107.3   33.3 

18.2 

29.6 

1.9. 

Other  problems  may  be  given  from  this  table,  particularly  those 
relating  to  the  area  and  population  of  the  pupils'  State  in  compari- 
son with  those  of  other  States. 

21  Ansivers:  2  Portugal,  138.3  England,  540.9  France, 
187.8  Germany,  236.7  China,  289.3  Italy,  261.8  Japan,  271.3 
Spain,  88.7.       4  $137238663     4919764.       5  951  h.     43/^  d. 


IV.  22]  teachers'  manual.  61 

23  Answers:  170200000.  2  ITO/a^g.  3  6122423  15251345 
115050  11597412  12G7G044.  4  G39148|a.  5  $3174  gain. 
6  51003321. 

23  Before  each  set  of  examples  and  problems  dealing  with  a 
new  fraction,  one  or  more  short  teaching  exercises  should  be  given 
in  Avhich  the  fraction  should  be  taught  both  by  itself  and  in  rela- 
tion to  fractions  already  known.  One  of  the  best  means  for  teach- 
ing fractions  is  the  disks  made  of  wood,  cardboard,  or  pasteboard. 
Those  for  the  teacher  might  be  six  or  eight  inches  in  diameter, 
and  those  for  pupils  two  or  three  inches  in  diameter.  If  they 
are  not  provided  by  the  school  authorities,  the  pupils,  doubtless, 
would  be  able  to  assist  the  teacher  in  cutting  them  from  old  boxes. 
Means  for  marking  and  cutting  should  also  be  provided.  If  the 
pupils  to  be  taught  are  few  in  number,  a  table  could  be  used  by 
the  teacher  for  placing  the  disks  in  teaching  the  various  operations ; 
in  case  a  large  number  have  to  be  taught  at  one  time,  the  teacher 
could  use  a  flat  surface,  inclined  in  such  a  way  as  to  permit  all 
the  pupils  to  see  the  disks  upon  it.  There  might  be  made  grooves 
upon  the  surface  to  permit  the  disks  to  remain  in  place.  In  teach- 
ing, sometimes  the  teacher  places  the  disks  for  the  pupils  to 
observe,  and  sometimes  the  pupils  place  the  disks  they  have  by 
direction  of  the  teacher.  The  small  disks  also  can  be  used  by  the 
pupils  in  preparation  of  a  lesson. 

The  preliminary  teaching  lesson  for  the  exercises  on  this  page 
is  indicated  by  the  following  questions,  which  are  asked  by  the 
teacher  as  he  places  the  disks  :  "  I  will  cut  this  circle  into  two 
equal  parts,  as  you  see.  What  is  this  part  called  (holding  up  one 
of  the  parts)  ?  this  part  (holding  up  the  other  part)  ?  How  many 
halves  in  one  ?  ^  and  ^  equals  what  ?  ^  taken  out  of  one  equals 
what  ?  If  I  take  the  half  two  times,  what  is  the  result  ?  How  many 
times  is  the  half  contained  in  the  whole  one  ?  Now,  to  review, 
i  +  i?  1  less  i?  2  times  i?  1  divided  by  i?"  The  pupils 
ought  to  be  ready  now  for  the  exercises  on  this  page.  If  they  find 
difhcvdty  with  the  problems  in  inches  and  half  inches,  similar  exer- 
cises with  measures  cut  from  cardboard  or  paper  might  be  given. 


62  GRADED   ARITHMETIC.  [IV.  24 

34—27  These  for  rapid  oral  work.  Do  not  leave  them  until 
the  pupils  are  able  to  perform  them  rapidly.  Before  9,  page  24, 
is  attempted,  a  short  teaching  exercise  should  be  given.  It  may 
be  necessary  to  perform  some  of  the  exercises  at  first  with  the  aid 
of  measures.  Other  exercises,  like  9,  page  25,  may  have  to  be 
illustrated  by  sticks  or  marks  ;  but  when  this  is  done  they  should 
be  reviewed  without  aids  of  any  kind. 

28—39  The  teaching  exercise  here  may  be  as  follows  :  "I 
will  cut  this  circle,  as  you  see,  in  4  equal  parts.  One  of  these 
parts  is  called  one  fourth.  What  is  this  part  called  (holding  up 
one  of  the  parts)?  What  is  this  part  called  (holding  up  another 
part)  ?  How  many  fourths  in  the  whole  circle  ?  How  many  fourths 
here  (pointing  to  two  parts  arranged  in  the  form  of  a  half -circle)? 
How  many  fourths  here  (pointing  to  three  parts)?  ^  plus  ^  plus  ^ 
plus  ^  are  what  (putting  the  parts  together  in  the  form  of  a  circle)  ? 
1  less  ^  is  what  (taking  away  one  of  the  parts  from  the  circle)? 
1  less  I  ?  1  less  f  ?  Hoav  many  times  have  I  taken  ^  (putting 
two  of  the  parts  together)?  2  times  ^  is  what?  How  many  times 
have  I  taken  ^  here  (putting  three  of  the  fourths  together)?  AVhat 
is  three  times  ^  ?  4  times  ^  ?  How  many  times  must  I  take  ^  to 
make  |  ?  to  make  f  ?  to  make  a  whole  one  ?  ^  is  contained  in  | 
how  many  times  ?  in  f  how  many  times  ?  in  a  whole  one  how 
many  times  ?  What  else  may  we  call  this  part  of  the  circle  (put- 
ting two  fourths  together)?  How  many  fourths  is  ^  ?  ^  less  ^  is 
what  ?  2  times  |-  is  what  ?  How  many  times  is  ^  contained  in  -^  ? 
^  of  |-  is  wliat  ?  "  Go  on  in  this  way  comparing  ^  with  f  and  1^, 
also  f  with  1^  and  1^. 

In  teaching  the  expression  of  the  fraction,  lead  the  pupils  to 
see  that  the  denominator  expresses  tlie  number  of  parts  into  wliich 
the  unit  is  divided,  and  the  numerator  the  number  of  parts  taken. 
Give  the  pupils  much  practice  in  this.  Finally,  they  should  be 
able  to  answer  the  questions  :  What  does  tlie  denominator  express  ? 
What  does  the  numerator  express  ?  They  sliould  also  be  able  to 
illustrate  by  objects  or  marks  their  answers. 

30   Tew  or  many  of  these  exercises  should  be  performed  by  the 


lY.  31]  TEACHEKS'    MANUAL.  68 

aid  of  disks  according  to  the  pupils'  understanding  of  them.  11, 
12,  and  13  can  best  be  performed  by  the  aid  of  sticks,  marks,  or 
dots.  Show  the  pupils  how  this  may  be  done.  After  these  have 
been  performed  objectively  the  pupils  might  be  led  to  give  little 
explanations  of  the  solution  thus  :  ":^  of  8  is  2;  f  of  8  is  3  times 
2,  or  6  ;  6  is  ^  of  2  times  6  ;  2  times  G  is  12.  2^  is  ^  of  4  times  2^; 
4  times  2  is  8,  and  4  times  i  is  1 ;  4  times  2^  is  9."  Explain  by 
examples  what  is  meant  by  ''  reducing  to  lowest  terms  "  as  required 
in  7. 

31  —  3*3  These  exercises  ought  to  be  performed  readily  without 
the  aid  of  objects  or  preliminary  questioning,  unless  in  some  of 
the  applied  problems  the  measures  are  not  familiar.  The  stej^s 
of  the  solution  should  be  given.  Thus,  in  9,  page  31 :  "^  from  ^ 
I  cannot  take  ;  so  I  take  ^  from  1^,  leaving  f.  I  took  1  from  8, 
leaving  7;  1  from  7  leaves  6.  Answer.  6f."  And  in  6,  page  32: 
"  The  horse  eats  3  times  ^  peck  or  1|-  pecks  in  1  day.  It  will  take 
as  many  days  for  him  to  eat  4^  pecks  as  1|-  is  contained  times 
in  4-^,  1-^  is  contained  in  4-^,  3  times.  Answer,  3  days.  Since  4|- 
bushels  is  4  times  as  much  as  4|-  pecks,  it  will  take  him  4  times 
3  days,  or  12  days."  Prol)ably  some  teaching  will  be  necessary 
for  this  problem,  but  the  pupils  should  be  permitted  to  try  to  solve 
the  problem  before  any  help  is  given.  If  they  solve  it  by  reducing 
the  pecks  to  quarts,  accept  the  solution. 

33  —  34:  For  teaching  eighths  and  their  relations  to  fourths 
and  halves  teachers  are  referred  to  the  note  in  which  a  method 
for  teaching  fourths  was  shown  (page  G2  of  Manual).  Essentially 
the  same  plan  should  be  pursued  here.  After  the  teacher  has 
taught  by  objects  all  the  important  facts,  short  dictation  exercises 
might  be  given  for  the  pupils  to  solve  with  disks  at  their  seats. 
Such  exercises  as  the  following  will  suggest  the  kind  of  work  which 
may  be  given  :  -g^  +  -g-  +  ^  equal  what  ?  -g  +  ^  ?  (Always  expect 
the  answer  to  be  given  in  the  simplest  form,  and  if  they  are  not 
so  given,  ask  questions  till  the  desired  answer  is  obtained.)  -g-  +  ^ 
+  i  +  i?  i  +  i?  i  +  i  +  i?  i  +  i?  i  +  i+i?  i  +  i  +  i? 
i  +  f?     f  +  i?     t  +  i?     f+i?     If  +  i?     If  +  f?      From  1 


64  GKADED    ARITHMETIC.  [IV.  35 

takef;  1  — f?  i-^?  f-i?  i— i?  i-f?  |  — i? 
f-f?  li-i?  li-f?  li-f?  Multiply  i  by  5 ;  fX2? 
fX2?  fX4?  fX3?  fX4?  |X2?  fX3?  fX4? 
fX6?  |X6?  iofi?  iofi?  ioff?  ioili?  iofli? 
How  many  times  is  ^  contained,  inf?     f-T--|-?     ^~r-  ^?     i~i~8'^ 

a_L.l?         1i_L.X?        5.-1.3? 
4'8*         -^¥-8-         4-F- 

The  first  18  exercises  on  page  34  ought  to  be  performed  first 
with  the  aid  and  afterwards  without  the  aid  of  the  cut  at  the  top 
of  the  page. 

35  —  36  Most  of  these  exercises  should  be  performed  without 
objects,  and  the  steps  of  the  solution  should  be  required,  as  in 
12,  page  35  :  "2  times  8  =  16;  f  of  8  =  3  ;  16  +  3  =19.  8  mul- 
tiplied by  2|-  =  19."  And  in  15  on  the  same  page:  "2^  =  ^; 
iofi  =  i;  iof  f  =  f;  f  off  =  V-orli" 

37—38  These  exercises  are  supposed  to  be  performed  orally; 
but  it  may  be  well  for  the  sake  of  clearness  and  the  acquirement 
of  a  good  form  of  analysis  for  future  use  to  have  the  pupils  write 
out  the  separate  steps  of  some  of  the  most  difl&cult,  as,  for  example, 
in  14,  page  37: 

Given  the  cost  of  f  bu.  apples. 
To  find  the  cost  of  2  bu. 

$0.60  =  cost  of  %  bu.  $1.60  =  cost  of  1  bu. 

0.20  =    "     ''  ^  bu.  3.20  =    "     "2  bu. 

9,  page  38.  60^  yd.  =  entire  piece. 

16i  yd.  X  3  =  48f  yd.  =  yards  cut  off. 
60^  yd.  —  48f  yd.  =  llf  yd.  remaining. 

39  Anstvers:  1  128^  65f  42f  58}  69|.  2  8i  43^ 
519f  105f  161^.  3  15^  189|  176^  868  996f  4  226f 
62f  179f  229i  118^.  5  131f  55^  50  259f  49f. 
6  4800  534f  163f  1050  712^.  7  16f  96f  58f  353f 
1353f.  8  60|  52i  251f  322  46f  9  2362^  49^  975f 
720  33.  10  152^  107f  95f  2766|  232.  11  30f  66} 
16  7875  1714^.  12  80^  27^  380  99^  72^.  13  456^  lb. 
14   $303f .         15   98  lots. 


IV.  40]  teachers'  manual.  65 

40  Answers :  1  82  books.  2  635^  bu.  3  $0.76.  4  30  mi. 
5  6^  mi.  6  22^  mi.  1676^  mi.  7  $267.75.  8  4  times  ; 
2/.  9  Horse,  $645  Carriage,  $215.  10  $1592^.  11  $128f. 
12  $7.56^.  13  $27.57f.  14  $254.00i.  15  $100.78+. 
16  $562.18f.  17  $311|.  18  7.60^.  19  $1.56^.  20  $1172^. 
21  $34.42f 

Good  forms  of  written  analysis  are  suggested  by  the  following : 

3         $0.15|-  selling  price  of  1  gal. 
42 


30 
60 

21^ 

$6.51 

5.75 

$0.76 


selling 

price 

of  42 

gal 

cost 

a 

a 

a 

gain. 

9     $860  =  cost  of  horse  and  carriage  =  4  times  cost  of  carriage. 
Cost  of  carriage  =  ^  of  $860  =  $215. 

41—42  One  or  more  teaching  exercises  should  precede  this 
work.  For  suggestions  as  to  how  they  may  be  conducted,  see 
directions  for  teaching  fourths  and  eighths.  As  soon  as  the  pupils 
can  discover  for  themselves  that  there  are  two  ways  of  multiplying 
a  fraction  by  a  whole  number  and  also  two  ways  of  dividing  by 
a  whole  number,  let  them  show  objectively  how  this  may  be  done, 
and  afterwards  state  the  facts  and  reasons.  For  example,  the 
pupils  should  be  led  to  show  that  |  can  be  multiplied  by  2  by 
increasing  the  number  of  parts  while  the  size  of  the  parts  remains 
the  same  or  by  increasing  the  size  of  the  parts  while  the  number 
of  parts  remains  the  same.  They  may  afterwards  make  the  state- 
ment that  a  fraction  may  be  multiplied  by  a  number  by  multiplying 
the  numerator  or  dividing  the  denominator  by  that  number. 

43  Exercises  for  practice,  which  should  be  performed  at  sight. 
If  the  pupils  cannot  perform  them  readily  at  first,  let  them  work 
for  a  while  with  the  disks.     This  will  be  good  desk-work. 


66  GHADED   ARITHMETIC.  [IV.  44 

44  Care  should  be  taken  to  make  the  work  in  division  very- 
simple  at  this  stage  of  the  pupils'  progress.  When  the  divisor  is 
a  fraction  it  should  be  contained  in  the  dividend  an  even  number 
of  times.  Lead  the  pupils  to  reduce  the  divisor  and  dividend  to 
the  same  denomination  before  dividing.  1  -i-  6,  f  -l-  3,  etc.,  in  4, 
should  be  solved  by  getting  ^,  ^,  etc.,  of  the  given  numbers. 

When  twelfths  are  taught,  let  the  pupils  have  much  practice 
in  finding  the  relation  of  twelfths  to  sixths,  thirds,  halves,  and 
fourths, 

45  Show  the  pupils  how  this  diagram  can  be  used  in  the  solu- 
tion of  the  problems,  and  give  similar  ones.  When  they  are  under- 
stood let  them  be  performed  without  aids  of  any  kind. 

46  The  work  in  division  here  may  be  too  difficult  without  some 
preliminary  teaching.  Besides  the  illustration  given  on  page  45 
for  teaching  division  by  a  fraction,  disks  or  squares  may  be  used 
in  which  the  divisor  and  dividend  are  made  to  be  of  the  same 
denomination.  Thus,  ^-i-  ^,  or  8§  -I- 1,  may  be  represented  ob- 
jectively as  the  division  of  sixths  by  sixths.  Until  the  pupils  are 
entirely  familiar  with  the  process  of  division  and  of  finding  the 
fractional  parts  of  numbers,  lead  them  to  illustrate  with  objects  or 
drawings  the  ojDcrations  called  for. 

47  Lead  the  pupils  to  state  reasons  in  their  own  language  first. 
Afterwards  they  can  be  led  to  improve  their  statements  by  some 
questions  or  suggestions  from  the  teacher  ;  or  diiferent  pupils  may 
be  called  upon  t©  perform  the  same  problem  to  see  who  will  give 
the  simplest  and  clearest  explanation.  By  substituting  whole 
numbers  for  fractions  the  conditions  of  a  problem  may  sometimes 
be  better  understood  ;  thus,  in  3,  if  the  pupil  hesitates  or  does  not 
know  whether  to  multiply  or  divide,  say  :  "At  $2  a  yard  how  many 
yards  of  cloth  can  I  buy  for  $8  ?  " 

49  In  all  cases  where  the  problem  can  be  solved  at  sight,  do 
not  permit  the  steps  to  be  given.  Answers  to  such  exercises  as, 
^-i-^,  4  -I-  -^j  and  -^^  X  2,  should  be  given  at  sight.  In  giving  the 
steps  of  a  solution  some  of  the  most  obvious  ones  should  be  omitted, 
as,  for  example,  in  adding  8^  and  4-j7^,  it  is  enough  for  the  pupil 


IV.  22]  teachers'  manual.  67 

to  say:  "y^^  cand  -/^  is  ll,  added  to  12  is  13^."  And  in  multi- 
plying IGy'V  by  8  to  say:  "S  times  16  is  128,  8  times  ^''(^  is  2f, 
added  to  128  is  130f ."  Long  and  labored  explanations  should  be 
avoided. 

50  A  table  that  can  be  used  for  drill  in  review  at  any  time. 
Other  exercises  than  those  indicated  can  be  given,  as  -E"  X  6,  C  X  12, 
/-^-■^,  etc.  In  denominate  numbers  the  table  can  be  used  in  many 
ways  and  to  an  almost  unlimited  extent.  But  in  using  drill  tables 
care  should  be  taken  not  to  practice  with  them  too  much.  On 
account  of  the  ease  with  which  drill  can  be  given  from  them,  the 
temptation  is  to  use  them  for  a  longer  time  than  is  really  needed 
for  facility  in  the  use  of  numbers. 

51—52  Kearly  all  of  these  problems  ought  to  be  performed 
orally;  but  for  the  purpose  of  learning  a  good  form  of  written 
analysis  the  pupils  should  be  asked  to  write  out  in  full  the  steps 
that  are  taken  in  the  solution  of  the  most  difficult  problems. 


53    Ajistvers:    1  ll^.       2  23^-       3 

ISf       4  2 

1| 

5  21§ 

6  34ii.        7  35§.        8  32.        9  42H- 

10  56*. 

11  103J^ 

12    24"8*.        13    388|f.        14    309f. 

15    610f|. 

16   272f 

17    281^7^.        18    376f.        19    348f. 

20    41l/. 

21    470f 

22  372f.       23  322i.       24  372|.       25 

424i. 

It  is  not  necessary  in  all  cases  to  write  out  in  full  the  solution 
of  these  problems,  but  only  such  parts  of  them  as  cannot  be  per- 
formed v/ithout  the  aid  of  figures.  Thus,  in  11  the  pupil  could 
say :  "  f  =  f ,  |  =  ^,  and  ^^  =  ^.  Thirds,  halves,  and  sixths  can  be 
reduced  to  twelfths.  j\  +  j%  ==  }|,  +  j%  =  }|,  +  j%  =  f §,  +  ^%  = 
ft  "^  -i-     -^  write  the  ^  and  add  the  two,  etc." 

54  A7iswers:  1  6^.  2  5^.  3  2f.  4  4^.  5  4^.  6  4^-^. 
7  8^.  8  r^i.  9  5^.  10  6f.  11  5f.  12  7^.  13  8f. 
14  6f.  15  13f.  16  7f.  17  10/^.  18  19|.  19  27}^. 
20  9|.  21  13/2.  22  I  23  lOi.  24  45f.  25  l^^. 
26  33f.  27  Of  28  62i.  29  12^^.  30  55^.  31  Sj\-. 
32  50 1 .   33  24f.   34  21^.   35  44^.   36  51f.   37  28|. 


68  GRADED   ARITIOIETIC.  [IV.  55 

38  66f.  39  159f.  40  86f.  41  141^^.  42  223^.  43  533^3^. 
44  607|.  45  220f.  46  188|.  47  2863.  48  377^72. 
49  319|.       50  389tV       51  163f.       52  353|.       53  652^3^. 

The  same  may  be  said  of  the  solution  of  these  problems  as  was 
said  of  the  solution  of  the  problems  on  the  last  page.  Require 
the  pupils  to  write  out  only  such  parts  of  the  solution  as  cannot 
be  performed  orally;  thus,  in  35  the  pupil  may  say,  as  he  performs 
the  problem  :  <'  |  from  ^  I  cannot  take  ;  |  from  1-^  or  y-  =  f  5  I 
write  the  |;  18  from  62,  44.  Answer,  44|."  And  in  41 :  "|  from  f. 
I  can  reduce  them  to  twelfths.     y\  from  ^^  =  -jL,  etc." 

55  Reasons  for  steps  taken  should  be  given  ;  thus,  in  2:  ''There 

are  4  fourths  in  1 ;  there  are  as  many  ones  in  190  fourths  as  four 

fourths  are  contained  times  in  190  fourths  or  47-^."     The  written 

«  ri.-         J.  n    J.  1-4  fourths)  190  fourths.     ^,  .     . 

term  of  solution  at  nrst  may  be  :  ^ This  is 

47^ 

a   good    time  to  look  upon  the  fraction  as  only  another  form  of 

division,  the  numerator  being  the  dividend  and  the  denominator 

being  the  divisor.     With  this  view  the  only  explanation  needed  is 

that  one  number  is  divided   by  another  number.     The  reduction 

of  mixed  numbers  to  fractional  numbers  may  be  explained  thus  : 

''There  are  ||  in  1;  in  8  there  are  8  times  }f  =  f|,  +  -jS^  =  -yji.." 

If  written,  the  form  may  be  simply  8^^^  =  f  f ,  +  y5_  =  j^o^l 

In  the  solution  of  the  exercises  in  multiplication  on  this  page 

write  out  only  such  parts  as  cannot  be  carried  easily  in  the  mind  ; 

thus,  in  33:  "3  times  2|  =  8i,  ^  of  2|  =  1/^;  8^  +  1^^  =  91^. 

The   written   form  which  appears   on  slate  or 

paper  is  2| 

X3i 

56  Answers:  llOSli.     2  1956f.    3  1455^.  8^ 
4  1615f.       5  I5873J5.       6  1398f.       7  2525f.  1^% 

8  1916i.     9  ISlOf.      10  2877/^.      11  2389f.  9}^    A7mver. 

12  6197H-        21  11^.        22  13^.        23  10§. 
24  12.       25  18.       26  21f.       27  41/^.      28  80^V      29  192^^ 
30  228.     31  346^.     32  339J.     33  177|.     34  219f.     35  145f. 
36  248^4.       37  79^^. 


IV.  57]  teachers'    MAiJUAL.  69 

Pursue  the  same  course  with  these  exercises  as  was  suggested 
for  previous  exercises.  When  necessary  the  divisor  and  dividend 
shouhl  be  reduced  to  the  same  denomination.  Instead  of  writing 
in  full  these  denominations,  the  pupils  may,  after  the  first  few 
exercises,  write  only  the  numerator  ;  thus  in  24  : 

3^42 

2       2    or  -8/  -f- 1  =  V  =  12  Answer. 

7~[84 

12  Answer. 

57  The  answers  to  some  of  the  exercises  contain  fractions 
having  denominators  larger  than  12  ;  but  it  is  understood  that  the 
divisor  and  dividend  when  reduced  to  the  same  denomination  may 
be  treated  as  whole  numbers  in  the  division. 

Answers:  1  -6-j<y>-  ^2°  \¥-  ¥/•  2  ^  2^\  3^%\.  3  13^ 
3/t-  Iff  4  lOfa  4^V4  ^2%%-  5  921f.  6  lUh  7  146. 
8  405.  9  880f  10  12  steps.  11  240  da.  12  39f  da. 
13    500  people.  14    365  da.         313  working  da.         $61^\. 

15  6m  "^i"-        16  i     2d  boy  earns  40/     3d  boy,  30/     4th  boy, 
30/.       17  $1440     11080     $1620. 

Eequire  the  pupils  to  make  drawings  of  the  solution  of  prob- 
lems whenever  it  is  possible  to  do  so,  and  to  Avrite  out  in  full  the 
steps  of  each  illustrated  sohition.  The  principle  involved  in  16 
is  supposed  to  have  been  taught,  but  it  may  require  some  leading 
to  get  the  pupils  to  illustrate  the  problem  with  dots  as  it  should  be. 
At  first  use  simple  denominations,  as  halves  or  thirds,  gradually 
leading  up  to  the  work  required.  The  finished  drawing  should 
represent  the  number  of  cents  that  each  boy  has  earned.  17  can 
be  performed  in  two  ways  :  to  first  find  the  value  of  the  whole  lot 
and  then  ^  of  it ;  or,  to  multiply  the  value  of  j\  of  the  lot  by  4, 
first  leading  the  pupils  to  see  the  ratio  of  -^  to  J^. 


58    Answers:   1  1717^  lb.     $4.29+. 

3  2411^  mi.    3375f  mi. 

4  7|.\.      5  $4,371.      6  13^  h.     14-  h. 

7  10  mi.     1^  h.       8  3  h. 

18f  mi.       9  195  mi.     2|  h. 

70  GEADED   ARITHMETIC.  [IV.  59 

Such  problems  as  9  can  be  illustrated  by  lines  on  the  board 
drawn  to  scale,  1  inch  to  a  mile,  placing  the  distance  above  the 
line,  and  time  below. 

59  Ansivers :  2  26  h.  15  min.  3  12  in.  entire  2f  in.  1  in. 
6  $5.16|. 

60-61  Stiff  paper  or  cardboard  should  be  used  for  these  exer- 
cises. Some  preliminary  exercises  may  have  to  be  given  to  lead 
the  pupils  to  see  clearly  the  relative  size  of  the  large  square,  the 
strips,  and  small  squares,  and  to  be  able  to  express  the  size  in 
decimals  as  required.  Such  exercises  as  the  following  may  be 
useful :  "  How  many  strips  in  the  large  square  ?  What  part  of 
the  large  square  is  each  strip  ?  2  strips  are  what  part  of  the  large 
square?     3  strips  are  what  part  of  the  large  square?     Hold  up 

4  tenths  of  the  large  square.  Hold  up  7  tenths  of  the  large  square. 
Hold  up  9  tenths  of  the  large  square.  Each  strip  is  how  divided  ? 
Each  small  square  is  Avhat  part  of  a  strip  ?  How  many  of  the 
small  sqiiares  are  there  in  the  large  square  ?  What  part  of  the 
large  square  is  each  small  square  ?  Cut  off  1  hundredth  of  the  large 
square.  1  more  hundredth.  How  many  hundredths  have  you  cut 
off  ?  Cut  off  as  many  small  squares  as  will  make  7  hundredths 
of  the  large  square.  Hold  up  1  tenth  and  2  hundredths  of  the 
large  square.  Hold  up  2  tenths  and  4  hundredths  of  the  large 
square.     Calling  the  large  square  one,  Avhat  may  we  call  1  strip  ? 

5  strips  ?  6  strips  ?  1  small  square  ?  3  small  squares  ?  7  small 
squares  ?  Hold  up  4  tenths  and  6  hundredths.  How  many  hun- 
dredths in  1  tenth  ?  in  3  tenths  ?  in  0  tenths  ?  How  many 
hundredths  in  2  tenths  and  3  hundredths  ?  in  4  tenths  and  7 
hundredths  ?  Write  1  tenth  in  the  common  form.  What  is  the 
denominator  ?  Another  way  of  writing  1  tenth  is  to  write  the 
numerator  1,  and  place  a  dot  called  a  decimal  point  before  it, 
thus,  .1.  A  fraction  whose  denominator  is  ten,  or  some  power  of 
ten,  is  called  a  decimal  fraction.  In  this  decimal  fraction  which 
we  have  written  what  only  is  expressed?  How  may  we  know 
that   the   figure   one  stands  for  1  tenth  ?     Kead  these  decimals  : 


rv.  62]  teachers'  isianual.  71 

.7 ;  .8 ;  .3.  How  should  we  express  10  tenths  ?  Hold  up  the 
part  of  1  which  these  fractions  express  :  .4  ;  .7  ;  .6.  When  we 
wish  to  express  hundredths  we  write  the  numerator  in  the  second 
place  to  the  right  of  the  decimal  point,  thus,  .04,  This  expresses 
what  fraction  ?  .38  expresses  how  many  tenths  and  how  many 
hundredths  ?     Kead  the  fraction  as  hundredths." 

After  such  an  exercise  the  pupils  should  be  ready  to  take  up 
the  work  given  on  these  pages.  The  last  five  exercises  of  page  61 
should  be  extended  until  the  reading  and  Avriting  of  decimals  to 
hundredths  are  thoroughly  understood. 

62  Thousandths  should  be  taught  in  a  way  similar  to  that  sug- 
gested above  for  teaching  hundredths.  Great  care  should  be  taken 
in  marking  and  cutting  the  squares  and  strips,  each  expression  of 
figures  representing  what  the  pupils  actually  see  to  be  the  frac- 
tional part  of  the  given  unit. 

63  Dwell  upon  each  step  here  given  until  the  pupils  thoroughly 
understand  it. 

64  In  4  lead  the  pupils  to  see  that  f  or  f  of  1  is  the  same  as  | 
or  I  of  jg§  or  |g§§.  f  of  100  hundredths  is  75  hundredths  ;  how 
expressed  ?  etc.  The  reduction  asked  for  in  5  should  be  performed 
as  simply  as  possible ;  for  example  :  "  250  thousandths  is  how 
many  hundredths  ?  25  hundredths  is  the  same  as  what  fraction  ?  " 
If  the  pupils  hesitate,  ask  what  50  hundredths  is  equal  to,  and 
then  25  hundredths.  In  6,  7,  and  8  lead  the  pupils  to  get  the 
required  fractional  part  of  100  hundredths  or  1000  thousandths  as 
before.  Give  other  exercises  similar  to  10,  that  the  principle  in- 
volved may  be  fully  understood. 

65  Answers:  10  54087.645.  11  376.848.  12  656.647. 
13  123.728.       14  407.503. 

The  first  four  exercises  are  important,  and  may  be  extended 
profitably.  Addition  of  decimals  ought  not  to  be  difficult  to  pupils 
who  have  been  carefully  trained  previously.  Let  the  explanations 
be  as  simply  given  as  was  advised  for  addition  of  whole  numbers. 
The  sum  24  thousandths  to  be  treated  as  hundredths,  and  thou- 


72  GRADED   ARITHMETIC.  [IV.  66 

sandths  ought  to  be  as  well  understood  as  so  many  tens,  to  be 
treated  as  hundreds  and  tens. 

66  Answers:  12  5.858.  13  .063.  14  15.731.  15  27.251 
74.066.  16  640.991  76.324.  17  65.147  78.591.  18  85.201 
86.308.       19  290.317     26.699. 

If  necessary  extend  the  objective  work  in  subtraction  of  deci- 
mals. Simple  explanations  of  steps  in  written  work  should  be 
made  by  the  pupils ;  thus,  in  15  :  "0  thousandths  from  1  thou- 
sandth is  1  thousandth  ;  1  is  equal  to  10  tenths,  and  1  tenth  is 
equal  to  10  hundredths  ;  5  hundredths  from  10  hundredths  is  5 
hundredths  ;  7  tenths  from  9  tenths  is  2  tenths ;  0  from  27  is  27. 
Answer,  27.251." 

67  Answers:   1  10.276  T.     2.525.      3  .495  T.      4  371.25  gr. 

5  27.4  gal.       6  79.255  A. 

If  necessary,  let  the  objects  be  used  in  multiplying. 

68  Answers:    1  .32.      2  .032.      3  .216.      4  1.184.      5  2.422. 

6  1.9.  7  6.75.  8  552.  9  1.218.  10  3.084.  11  5.715. 
12  5.201.  13  5.272.  14  6.68.  15  5.022.  16  9.048. 
17  13.344.  18  10.89.  19  18.252.  20  20.925.  21  25.05. 
22  39.088.  23  73.104.  24  102.942.  25  151.552.  26  255.948. 
27  268.584.  28  405.405.  29  691.01.  30  5918.4.  31  1168.50. 
32  2577.748.  33  46840.8.  34  2138.752.  35  29238.48. 
36  1762.592.  37  4350.675.  38  1865.528.  39  5659.316. 
40  3645.32. 

69  Answers:  1  5049.156.  2  4750.395.  3  2200.44. 
4  6796.998.    5  7315.934. 

Answers  to  nearly  all  the  remaining  problems  should  be  written 
without  any  figures  of  solution.  In  all  cases  where  the  multiplier 
is  50  or  25,  first  multiply  by  100  by  removing  the  decimal  point 
two  places  to  the  right,  and  divide  by  2  or  4. 

70-71  Dividing  any  number  by  an  integer  is  simply  getting 
a  fractional  part  of  the  number,  the  product  being  of  the  same 
denomination    as    the    multiplicand.      The    so-called    process   of 


IV.  72]  TEACHEKS'    MANUAL.  73 

"  niultipl^'ing  by  a  fraction  "  is  also  getting  the  fractional  part  of  a 
number.  Thus,  8.4  X  .9  may  be  read  9  tenths  of  8.4,  If  this  is 
not  clearly  understood  let  the  pupils  use  objects  as  indicated. 

73  In  every  operation  with  the  objects  let  the  process  and 
result  be  represented  by  figures.  Pupils  will  readily  see  in  work- 
ing with  objects  that  when  the  divisor  is  a  fraction  both  dividend 
and  divisor  are  to  be  of  the  same  denomination  before  dividing. 

73  Problems  in  which  large  numbers  are  used,  as  in  10,  19, 
20,  22,  need  not  be  solved  objectively.  If  the  divisor  is  an  integer 
it  will  be  sufficient  to  get  the  fractional  part  of  the  number  reduced 
to  tenths,  hundredths,  or  thousandths,  and  if  the  divisor  is  a  frac- 
tion, to  reduce  both  dividend  and  divisor  to  the  same  denomina- 
tion ;  thus,  in  10  the  pupils  may  say  4  thousandths  in  60000 
thousandths  15000  times.  And  in  22  :  ^l^  of  120  tenths  is  1 
tenth.     These  solutions  would  be  written  as  follows  : 

.004)60.000  '  120)12.0 

15000. 

74  Answers:  1  6.1  3010  3010  16.5 
340  200  .09.  3  755  600.1  101.5  50 
2  400  .9.  5  3.2  .18  160.  17.8. 
120.48  .438  2166.4  9556.  7  11760 
27.738.  8  142.048  2360  27  3.2  3002. 
.979  2890  39.606.  10  244.664  175 
11  13.635     792     249.66     9.1224     1000. 

The  pupils  see  in  multiplying  that  the  multiplication  of  tenths 
by  units  gives  tenths,  that  tenths  of  tenths  are  hundredths,  and 
that  tenths  of  hundredths  are  thousandths  ;  and  from  these  facts 
are  able  to  make  and  follow  a  simple  rule  for  pointing  oft".  In  the 
same  way  rules  made  for  pointing  off  in  division  may  be  followed, 
but  in  all  cases  the  pupils  should  be  ready  to  give  reasons  for 
pointing  off  as  they  do.  12  and  19  should  be  performed  by  re- 
moving the  decimal  point  to  the  left.  Answers  only  of  such 
problems  need  be  written. 


1 

403.5. 

2  2. 

46     780 

2.5. 

4   .04     25.5 

.15. 

6 

1541.5 

1010 

87.5 

10.25 

9  23.008 

120.355 

3.772 

325 

193.04. 

74  GKADED   AEITHMETIC.  [IV.  75 

75  These  exercises  should  be  performed  orally.  To  divide  by 
50  or  25  divide  by  100  and  multiply  the  quotient  by  2  or  4. 

76  Anstvers:  1  110  yd.  1.1  yd.  2  20  paces.  3  $3  15/. 
4  20/  $1.28.  5  20.  6  5.  7  72.  8  $15.  9  $1300. 
10  $55.04.  11  2000  pencils.  12  $36000  $44000  $4800. 
13  4.08.  14  792  &Q,.  15  792  ft.  107.25  ft.  303.60  ft. 
16  4  rd.     86  rd. 

These  problems  should  be  performed  without  much  figure  work, 
unless  the  steps  of  the  solution  are  required. 

77  After  distances  are  known  by  measurement,  they  should  be 
placed  upon  the  board  and  kept  there  for  convenience  of  com- 
parison. Other  known  distances,  such  as  the  distance  from  one 
town  or  city  to  another,  might  also  be  posted  for  reference. 

78  Answers:  7  2  mi.  1  mi.  2  mi.  4  mi.  3  mi.  40  rd. 
8  275  ft.  17160  ft.  31680  ft.  74.25  ft.  9  83^  ft.  204.75  ft. 
9240  ft.  235^  ft.  10  413|  yd.  6600  yd.  20  yd.  11  1||  mi. 
l^V^  mi.     17^  mi.     14f  |  mi.      12  61-^  ft.     20^  yd.     $2.56. 

Most  of  these  problems  can  be  performed  orally  by  the  pupils. 
Steps  and  reasons  for  steps  should  be  called  for  occasionally. 
After  a  problem  is  jjerformed,  and  the  correct  answer  given, 
questions  might  be  asked ;  as  for  example  in  12  :  "■  What  do  you 
find  first  ?  How  do  you  find  it  ?  How  reduce  to  yards  ?  Why  ? 
How  do  you  find  the  cost  ?  Why  ?  "  Generally,  however,  a  full 
statement  of  the  process  should  be  given  by  the  pupil  without 
interru2:)tion. 

79  In  finding  the  square  contents  of  a  surface,  lead  the  pupils 
to  think  first  of  the  number  of  units  in  a  row,  and  to  multiply  this 
number  by  the  number  of  rows,  as  was  shown  in  Book  III. 

80  If  the  pupils  have  not  had  previous  practice  in  drawing  to 
scale  in  connection  with  other  studies,  some  time  may  well  be 
spent  upon  it  here.  These  exercises  will  suggest  other  work  of 
the  same  kind  which  may  be  given.  An  explanation  of  10  might 
be  somewhat  as  follows  :  "  There  are  12  ft.  in  144  in.  and  8  ft.  in 


rv.  81]  teachers'  manual.  75 

96  in.  In  a  surface  12  ft.  long  and  1  ft.  wide  there  are  12  sq.  ft. 
In  a  surface  12  ft.  long  and  8  ft.  wide  there  are  8  times  12  sq.  ft., 
or  96  sq.  ft." 

81  In  such  problems  as  1,  5,  and  6,  lead  the  pupils  first  to 
reduce  the  dimensions  to  like  terms  ;  thus  :  "  20  yd.  =  60  ft. 
60  sq.  ft.  X  20  =  1200  sq.  ft."  This  problem  would  be  explained 
as  previously  shoAvn.  In  7,  let  the  pupils  find  first  the  number  of 
square  yards  that  will  exactly  cover  the  seat,  and  afterwards  the 
amount  which  would  be  required  if  in  covering  there  were  a  waste 
of  6  in.  in  the  length  and  width.  In  9,  let  as  many  pupils  as  can, 
make  the  proper  allowance  for  corners  in  finding  the  area  of  the 
walk.     Kequire  plan  to  be  drawn, 

83  Answers:  1  320  steps.  2  62  trees.  3  30375  sq.  ft. 
3375  sq.  yd.  4  5103  sq.  ft.  5  351  sq.  yd.  6  170  sq.  ft. 
57  ft.  7  238941  sq.  ft.  625|  ft.  8  $3361.16  $32.77. 
9    5|  rd.     10  27^^  mi.     86^  mi.     64f  mi.     11  25  sq.  yd.     25  yd. 

Besides  drawing  a  plan  for  the  solution  of  each  of  these  problems, 
the  pupils  should  be  expected  to  write  out  a  brief  analysis.  The 
following  solution  of  the  last  part  of  8  may  serve  as  an  example 
of  what  may  be  done  : 

178i  ft.,  length  of  lot.  $0.20    cost  of  1  yd. 

2_  163| 

357    ft.,  length  of  two  sides.  $3260 

134^  ft.,  width  of  lot.  16f 


3)  491^  ft.,  length  of  two  sides  and  end.         $32.76f  cost  of  fence.- 
163|  yd.,  length  of  fence. 

83  Answers:  1  13^  sq.ft.  2  6i  sq.  ft.  12^  sq.ft.  8  lO,?^  in.. 
9  41f  ft.  10  15  ft.  11  90  ft.  12  128.125  sq.  ft.  432.25 
sq.  ft.     14.2361  sq.  yd. 

84  A  teaching  exercise  upon  angles  and  right  angles  should 
precede  these  exercises.  If  the  teacher's  definition  of  angle  is  the 
difference  of  direction  of  two  lines,  he  places  upon  the  board  pairs 
of  lines  extending  in  different  directions,  and  shows  that  the  angles 


76  GRADED   AEITHIMETIC.  [IV.  85 

are  of  different  sizes.  In  all  these  examples  the  lines  forming  the 
angles  should  be  represented  as  meeting,  and  the  point  where  the 
lines  of  the  angle  meet  is  called  the  vertex.  Proceeding,  the  teacher 
draws  two  lines  crossing  each  other,  thus,  +,  and  says,  when  two 
lines  cross  each  other  so  as  to  make  four  equal  angles,  the  angles 
are  called  right  angles.  The  pupils  are  led  to  describe  an  angle  as 
the  difference  of  direction  of  two  lines,  and  a  right  angle  as  a  square 
corner.  Angles  similar  to  those  in  1  are  drawn,  and  the  pupils 
are  led  by  questioning  to  say  that  an  obtuse  angle  is  an  angle 
greater  than  a  right  angle,  and  that  an  acute  angle  is  an  angle  less 
than  a  right  angle. 

Before  parallelogram  is  taught,  parallel  lines  are  drawn,  and 
described  as  lines  having  the  same  direction,  or,  as  lines  which 
are  as  far  apart  in  one  place  as  in  another,  or,  as  being  the  same 
distance  apart.  Then  a  figure  is  drawn  whose  opposite  sides  are 
parallel,  and  the  figure  is  described  as  it  was  drawn,  the  descrip- 
tion being  drawn  from  the  pupils  by  questioning.  The  terms  base 
and  altitude  are  also  taught,  and  the  pupils  are  led  to  point  out 
the  base  and  altitude  of  several  parallelograms.  Parallelograms 
with  right  angles  and  those  that  are  not  so  formed  should  be 
drawn,  and  the  pupils  should  be  told  that  parallelograms  with 
right  angles  or  square  corners  are  rectangles.  They  should  be 
led  to  see  that  all  figures  whose  square  contents  they  have  thus 
far  found  are  rectangles.  They  are  now  ready  for  the  work  desig- 
nated in  2  and  3,  which  should  be  repeated  until  they  see  that 
the  area  of  any  parallelogram  can  be  found  by  niultij)lying  the 
length  by  the  width,  or  the  base  by  the  altitude. 

85  Applications  of  the  principle  developed  in  the  last  lesson 
should  be  made  \intil  the  pupils  can  readily  find  the  area  of  any 
parallelogram.  The  "diamond"  used  in  playing  baseball  is  a  square 
whose  edge  is  90  ft. 

As  the  pupils  proceed  to  find  the  relative  size  of  a  parallelogram 
and  triangle  of  the  same  base  and  altitude,  do  not  assist  them. 
Simply  ask  the  question  at  first,  and  give  directions  for  finding 
the  answer  only  as  they  are  found  to  be  necessary.     The  statement 


rv.  86]  teachers'  ]\IANUAL.  77 

in  answer  to  the  question  contained  in  5  should  be  made  by  the 
pupils  themselves. 

86  Answers :  4  26400  sq.  ft.  5  8  ft.  6  ft.  6  ft.  4  ft. 
36  sq.  ft.  6  9  ft.  7  180  sq.  ft.  20  sq.  yd.  8  20  sq.  yd. 
9  80  yd. 

No  assistance  should  be  given  the  pupils  beyond  what  is  given 
in  1.  It  would  be  well  to  give  several  problems  in  finding  the 
area  of  triangles.  It  will  be  profitable  and  agreeable  to  the  pupils 
to  give  them  problems  similar  to  5.  Give  measurements  of  such 
figures  by  dictation ;  say  that  the  figures  are  drawn  to  a  given 
scale,  and  ask  for  the  area.  The  following  dictation  exercise  may 
serve  as  a  model :  "  Draw  a  line  AB  1  inch  long.  From  the  point 
A,  perpendicular  to  AB,  draw  a  line  3  inches  long.  Mark  the  end 
of  this  line  C.  Join  BC.  From  a  point  D  in  the  line  AC,  2  inches 
from  A,  draw  a  line  parallel  to  AB.  Mark  the  end  of  line  last 
drawn  JS.  Supposing  these  lines  to  be  drawn  to  a  scale  of  12  rods 
to  an  inch,  find  the  area  of  each  figure  drawn."  Similar  figures 
drawn  on  the  board,  representing  fields,  gardens,  etc.,  will  give 
interesting  work  to  pupils. 

87  ^/m^ers.-  2  $16.80.  3  224  ft.  1536  sq.  ft.  107|  sq.  yd. 
4    6  sq.  ft.         5.    108  sq.  ft.       162  sq.  ft. 

A  teaching  exercise  may  be  necessary  before  the  pupils  can  find 
the  areas  of  a,  b,  and  h.  It  may  be  sufficient  to  ask  them  to  make 
light  dotted  lines  so  as  to  make  each  of  the  figures  to  consist  of 
a  rectangle  and  a  right  triangle.  Work  similar  to  1  may  be  given 
both  as  drill  and  test  exercises.  Give  as  little  direct  assistance  as 
possible  to  the  pupils  in  finding  these  areas.  In  addition  to  plans 
the  pupils  should  write  out  in  full  the  solution  of  the  last  four 
exercises. 

88  Ansivers:  1  15  ft.  2  3600  sq.  ft.  3  27^  sq.  yd. 
4  60  sq.  yd.  $14.40.  5  128^  sq.  ft.  6  22  sq.  ft.  7  9i  sq.  ft. 
8  22500  sq.  ft. 

The  pupils  should  draw  a  plan  illustrating  the  shape  and  loca- 
tion of  the  lot  upon  the  street,  in  the  solution  of  8.    Let  the  pupils 


78  GRADED   ARITHMETIC.  [TV.  89 

be  free  to  choose  the  conditions  of  required  original  problems. 
Encourage  them  to  give  as  difficult  problems  as  they  are  able  to 
solve. 

89  Show  to  the  pupils  as  many  of  these  coins  as  can  be  ob- 
tained. Of  the  coins  mentioned  the  gold  three-dollar  piece,  the 
gold  dollar,  and  the  silver  half-dime  are  not  now  coined.  The 
silver  three-cent  piece  also  is  not  coined.  The  number  of  cents  in 
fourths,  eighths,  fifths,  thirds,  and  sixths  of  a  dollar,  should  be 
remembered,  and  problems  involving  these  sums  should  be  per- 
formed orally. 

90  Tliis  account  should  be  copied,  and  each  entry  explained 
before  the  account  of  the  following  week  is  written.  Balance  on 
hand  in  2,  $2.55;  in  3,  $111.89. 

91  Require  the  pupils  to  make  the  ruling  for  a  bill  ;  also  to 
copy  and  finish  the  bill  given  on  this  page.  Explain  tlie  form  of 
dating  and  receipting  bills.     The  amounts  of  bills  are  as  follows  : 

1  .$17.71.  2  $1338.70.  3  $413.50.  4  1294.50.  5  $69.46. 
6  $1801.75. 

92  A?isi(fers:  1  $1444.80.  2  $4015.92.  3  $3465.60. 
4  $464.75     $302.40.       5  $21.87.      7  25s.     72s.       8  4     32     25. 

9  £3    £4^.     10  138s.    175s.     11  5  sov.     12  $.243.     13  $1,458 

$3,645     $7.29     $17,496     $42,525     $59,778     $10.8135. 

As  many  of  the  English  coins  as  can  be  obtained  should  be 
brought  into  the  class,  and  their  value  given.  Show  the  greater  con- 
venience of  our  decimal  system  both  in  writing  and  in  reckoning, 

93  Most  of  these  problems  can  be  performed  orally  by  the 
pupils.  Reasons  for  multiplying  or  dividing  in  reduction  may  or 
may  not  be  given. 

94  Answers:    7  10  lb.  5  oz.        8  15  lb.  6  oz.       9  26  lb.  11  oz. 

10  18  lb.  7  oz.  11  7  cwt.  50  lb.  12  11  cwt.  13  4  cwt.  20  lb. 
14  8  cwt.  70  lb.  15  4  T.  800  lb.  16  10  T.  17  9  T.  100  lb. 
18  17  T.  300  lb.  19  2  lb.  10  oz.  20  1  lb.  12  oz.  21  1  lb.  9  oz. 
22  4  lb.  6  oz.  23  1700  lb.       12  cwt.       600  lb.         24  14  cwt. 


IV.  95]  teachers'  manual.  79 

1600  lb.  25  1  T.  16  cwt.  26  2  T.  14  cwt.  27  4  T.  11  cwt. 
28    1  lb.  29    1  lb.  8  oz.  30    1  cwt.  31    2  cwt.  40  lb. 

32   1  T.  4  cwt. 

As  many  as  possible  of  these  problems  should  be  performed 
orally,  but  for  the  sake  of  exercise  in  analysis  the  solution  may 
be  written.  Explanations  should  be  the  same  as  in  addition  and 
subtraction  of  simple  numbers. 

95  A7isivers :    5    25  lb.  8  oz.         6    14  lb.         7    13  lb.  11  oz. 

8  11  lb.  4  oz.     9  66  lb.  12  oz.      10  43  lb.  5  oz.      11  118  lb.  2  oz. 

12  142  lb.  8  oz.  13  13  T.  14  33  T.  4  cwt.  15  15  T.  1800  lb. 
16  33  T.  4  cwt.  17  37  T.  10  cwt.  60  lb.  18  37  T.  12  cwt.  64  lb. 
19  95  T.  11  cwt.  20  lb.    22  4  lb.  8  oz.    23  4  lb.  8  oz. 

24  2  lb.  lOf  oz.  li  oz.   3  lb.  12  oz.   2  T.  5  cwt.  3  cwt.  60  lb. 

25  2  T.  6  cwt.  2  T.  13^  cwt.  1  T.  5  cwt.  10  T.  17  cwt.  50  lb. 
30  $120.       31  $69. 

Explanations  may  be  as  in  simple  numbers  ;  thus,  in  9  :  "12  times 

9  oz.  =  108  oz.  =  6  lb.  12  oz.  I  write  the  12  oz.  and  add  the  6  lb. 
to  the  next  product.  12  times  5  lb.  =  60  lb.  +  6  lb.  =  66  lb.  Answer, 
66  lb.  12  oz."  And  in  24  :  "  ^  of  24  lb.  =  2  lb.,  and  6  lb.  remainder 
=  96  oz.     I  of  96  oz.  =  lOf  or  lOf  oz.     Anstver,  2  lb.  lOf  oz." 

96  These  are  oral  exercises,  which  can  also  be  given  for  desk- 
work,  to  be  brought  into  the  class.  Explanations  of  processes 
might  be  given  in  the  recitation. 

97  Anstvers:  1  5  bu.  3  pk.  2  6  bu.  0  pk.  3  11  bu.  1  pk. 
4  8  bu.  2  pk.  1  qt.  5  12  gal.  6  13  gal.  3  qt.  7  11  gal.  2  qt.  1  pt. 
8  3  gal.  3  qt.  1  pt.  1  gi.         9  11  gal.  1  qt.  0  pt.  3  gi.        10  6  bu. 

2  pk.  6  qt.  11   7  bu.  2  pk.  5  qt.         12  1  bu.  1  pk.  3  qt.  1|-  pt. 

13  1  pk.      2  pk.  4  qt.      3  pk.  6  qt.        14  2  bu.  2  pk.         15  2  bu. 

3  pk.  16  2  bu.  2  pk.  17  3  pk.  6  qt.  18  4  gal.  3  qt. 
19  3  gal.  3  qt.  20  5  gal.  1  qt.  21  4  gal.  2  qt.  1  pt. 
22  1  gal.  1  qt.  3  gal.  1  qt.  23  3  pk.  4  qt.  3  pk.  5  qt. 
24    1  pk.  1  pt.           25    3  gal.  1  pt.  1^  gi.         3  gal.  1  qt.  i  pt. 


80  '  GRADED   ARITHMETIC.  [TV.  98 

26    8  gal.  27    7  qt.  28    7  pk.  4  qt.  29    5  bu.  2  qt. 

30   25  gal.  2  qt. 

These  may  be  given  for  written  desk-work,  although  the  pupils 
ought  to  be  able  to  perform  them  orally. 

98  Answers :    1    13  bu.  2    34  gal.  2  qt.         3    80  bu.  1  pk. 

4  21  gal.  5  153  gal.  3  qt.  6  99  bu.  7  61  bu.  8  152  gal. 
2  qt.  9  1  pk.  2  qt.  1^  pt.  10  2  qt.  1  pt.  1^  gi. 
11  5  bu.  3  pk.  12  7  gal.  |  qt.  13  8  gal.  1  qt.  14  2  bu.  1  pk. 
15  7  pk.  4f  qt.  16  2  bu.  25 1  qt.  17  2  gal.  1^  qt.  18  9  qt. 
1  pt.  2i  gi. 

For  variety  of  form  the  pupils  might  be  led  to  say  in  such  prob- 
lems as  21 :  "  If  I  give  to  one  person  3  pecks,  I  can  give  3  bushels 
or  12  pecks  to  as  many  persons  as  there  are  3  pecks  in  12  pecks. 
There  are  four  3  pecks  in  12  pecks.  So  I  can  give  12  pecks  to  4 
persons."    But  this  or  any  one  form  should  not  be  insisted  upon. 

99  Answers:     1    $36.  2     $15.50.         3    $8.         4    $1.92. 

5  $11.71|.  6  $3.53^.  7  5  bu.  3  pk.  6H  qt.  8  90/. 
9  160  sec.  405  sec.  10  40  min.  45  min.  210  min.  11  18  h. 
9  h.     117  h.       14  3  h.  30  min. 

100—101  These  problems,  which  may  be  recited  orally  in 
recitation,  might  be  given  for  desk-work.  In  finding  the  differ- 
ence of  time  in  years,  months,  and  days,  let  the  time  be  given  first 
in  entire  years  or  months  ;  for  example,  in  1,  page  101  :  "  Erom 
Feb.  22,  1732,  to  Feb.  22,  1799,  it  is  67  years.  From  Feb.  22  to 
Nov.  22  it  is  9  months.  From  Nov.  22  to  Dec.  14  it  is  8  + 14  days 
or  22  days.     Ansiver,  67  years,  9  months,  22  days." 

103  Answers:  1  'i  v.  20  r.  26|  q.  375  q.  10  r.  '2  48  r. 
500  q.  120  sheets.  3086f  sheets.  3  98  r.  15  q.  4  500  r. 
5  480  q.  6  24  r.  7  $17.50.  8  $1795.32-*.  9  15  doz. 
4f  doz.  10  44  doz.  105  doz.  11  36/.  12  24  boxes. 
13  $90. 

103    Answers:    10  $3305.76.      11  $10374.67.      12  $2283.85. 
If   answers  to  exercises  from  1  to  9  cannot  be  quite  readily 
given,  drill  the  pupils  upon  the  exercises  on  page  1. 


IV.  104]        teachers'  manual.  81 

104  Answers:  1  82.  2  75.  3  82.  4  77.  5  78. 
6  75.  7  73.  8  78.  9  70.  10  82.  11  51.  12  47. 
13  72.  14  57.  15  50.  16  G3.  17  62.  18  49.  19  45. 
20  49.  21  49.  22  52.  23  55.  24  67.  25  697.  26  597. 
27  669.  28  657.  29  639.  30  543.  31  539.  32  620. 
33  408.  34  560.  35  464.  36  523.  37  438.  38  392. 
39  389.   40  388.   41  455.   42  463.  43  505. 

Pupils  should  be  required  to  add  by  lines  as  they  are,  and  not 
write  the  numbers  in  columns  before  adding. 

105  A7istvers:  1  1800  bbl.  2  $4412657.81+.  3  $1272.90. 
4  $112.       5  $346.18.       6  $222.61. 

106  Atistvers:     7    $1.38.        8    150   overcoats.         9    73f  bu. 

10  $214.08.       11  74  yd.     $68.45. 

107  Answers:     9     $14,024-       $4488.  10     $3.24       $135. 

11  $10.33i     $121.66|.       12  $16.80     21  da.       13  330  paces. 

After  working  through  the  unit,  as  the  pupils  should  be  expected 
to  at  first,  they  might  afterwards  be  led  to  get  the  desired  result 
directly,  as,  for  example,  in  3  :  "It  will  take  60  men  ^  as  long  to 
do  the  work  as  it  takes  20  men.     -^  of  180  days  is  60  days." 

108  Answers:  1  3  yr.  4  mo.  5  yr.  6  mo.  2  2112  yr.  3  8^ 
S?>i    330.         4  7|f  oz.         5  $19.98.         6  25  wk.         7  Friday. 

8  66  doz.       9  24/.       10  $2240. 

109  Amwers:  3  $299.25.  4  $11.70.  5  C.  Sea  42900  sq. 
mi.  larger.  6  3753  ft.  higher.  7  $1.50.  8  76^%.  9  $360. 
10  7.2  mi.     8^  min. 

110  Ansivers:    1  $94.50.       2  40  mi.     5  h.  45  min.      3  $2.40. 

4  $84560.       7  3576  sq.  ft.       8  1192  pickets.       9  133^  sq.  yd. 

111  Answers:   1  134^  sq.  ft.     2  $850.     3  $3312.      4  200  yd. 

5  456  ft.      6  $573.60.      7  18  min.     121^  min.      8  1904213||  T. 

9  6748  poles.         10  10000  cocoons     3409jV  mi. 


82  GRADED   ARITHMETIC.  [V.  1 

SECTION   VII. 

NOTES   FOR   BOOK   NUMBER   FIVE. 

Before  taking  up  the  work  embraced  in  this  book,  the  pupils 
are  supposed  to  have  a  thorough  knowledge  of  common  fractions 
to  twelfths  and  of  decimal  fractions  to  thousandths.  They  are 
also  supposed  to  have  had  considerable  practice  in  finding  the  areas 
of  parallelograms  and  triangles,  and  in  performing  problems  in- 
volving the  common  weights  and  measures.  Teachers  are  advised 
to  look  over  Book  IV.  to  see  if  some  parts  of  that  book  may  not 
be  reviewed  before  taking  up  the  work  of  this  book.  Teachers 
are  also  referred  to  the  Note  to  Teachers  of  Book  V.  for  general 
suggestions. 

1—  2  There  should  be  practice  upon  these  exercises  until  answers 
are  given  with  a  good  degree  of  promptness.  At  first  it  may  be 
necessary  to  solve  some  of  them  by  steps  ;  for  example,  in  3,  page  2, 
the  pupils  may  be  led  to  say,  in  multiplying  47  by  8  :  "  320,  56, 
376."  Or  in  14,  page  1 :  "400,  85,  485  ;  300,  135,  435."  But  by 
degrees  answers  should  be  given  at  sight,  or  the  steps  should  be  so 
quickly  taken  as  to  make  the  exercises  practically  sight  exercises. 
Previous  practice  should  have  made  the  pupils  familiar  with  the 
products  of  all  numbers  to  20  by  all  numbers  to  10,  i.e.,  136  should 
be  recognized  as  the  product  of  17  and  8  as  readily  as  96  is 
recognized  as  the  product  of  12  and  8.  The  reverse  operations 
in  division  should  be  equally  familiar. 

Some  analysis  may  also  be  necessary  in  the  solution  of  the  exer- 
cises in  fractions,  but  such  analysis  should  be  as  limited  as  pos- 
sible ;  for  example,  in  19,  page  2,  the  pupil  may  think  and  say  '■'  |, 
h  f '  "^f-"  After  a  while  a  less  number  of  steps  than  are  here  given 
will  be  needed. 

3i-G  Short  and  naturally-expressed  explanations  of  these  exer- 
cises should  be  made  by  the  pupils.     The  following  will  serve  as 


V.  7]  TEACHEES'    MANUAL.  83 

examples  of  what  might  be  expected  of  them :  26,  page  3 :  "  62^/  = 
f  of  a  dollar.  10  bushels  will  cost  10  times  f  of  a  dollar  or  $  \°-. 
As  many  dozen  eggs  will  be  worth  this  as  $f  is  contained  times  in 
$Y-,  OJ"  l<5f  dozen."  18,  page  4:  ''3  rods  =  49^  ft.  A  surface 
49^  ft.  long  and  1  ft.  wide  will  contain  49^  sq.  ft.  A  surface  of 
the  same  length  6  feet  wide  will  contain  6  times  49^  sq.  ft.  = 
240  +  54  +  3  =  297  sq.  ft."  16,  page  5  :  "30  apples  will  cost  10 
times  as  much  as  3  apples.  10  times  2  cents  =  20  cents,  cost. 
The  boy  wovdd  get  15  times  as  much  for  30  apples  as  he  gets 
for  2  apples.  15X3  cents  =  45  cents.  He  gained  45  cents  less 
20  cents,  or  25  cents." 

7  Answers:  1  $20415.16.  2  $30446.15.  3  $23953.38. 
4  $6807.89.  5  $2685.25.  6  $543.75.  7  $278.64.  8  $476.28. 
9  $980.  10  $14.72.  11  $216.  12  $94.50.  13  $22.68. 
14  $31.65^.  15  $118.71.  16  $294.  17  $2033^,  son;  $18300, 
widow.       18  $2218|. 

Let  the  pupils  understand  that  in  all  final  results  of  cost  or 
selling  price,  when  the  fraction  of  a  cent  is  less  than  half,  the 
fraction  is  dropped,  and  when  the  fraction  is  one  half  or  over  one 
half,  an  extra  cent  is  added. 

8  Answers:  1  Total  area,  51250800  sq.  mi.;  Total  population, 
1467920000.  Population  per  sq.  mi.:  Europe,  101.3;  Asia,  57.7; 
Africa,  11.03 ;  Australasia  and  Pacific,  1.4 ;  N.  A.  and  W.  I.,  13.8 ; 
So.  A.,  5.3  ;  Polar  regions,  .06.  5  $11.25  $15  $10.35.  6  $117 
$175.50  $130.  7  $57  $49.59  $42.75  8  $50.62^  $70.87^ 
$54.  9  $224i  $243f  $260.  10  $4009.50  $3766.50  $3098.25. 
11  $14700  $21840  $28560.  12  $54  $66  $32.  13  $180 
pencils     900  pencils     2250  pencils. 

Several  problems  may  be  made  of  2,  3,  and  4.  Let  the  pupils 
make  as  many  as  they  can  before  the  teacher  suggests  any. 

9  A7iswers :  1  $4000  profit.  3  1600  oranges.  4  127f  bu. 
$76.65.       5  89^^  bbl.       6  $93.       7  14452.1.       8  4435.33. 


84  GRADED   AEITmHETIC.  [V.  lO 

9  13960.22.  10  111.607.  11  5604.935.  12  479.917. 

13  6066.925.  14  28608  123586.56  288654.72.  15  100  10  1. 
16  1600  240  32.  17  1710  4788  4959.  18  6.88  54544 
35.088.       19  5407.2    48.064    3652.864.       20  7.68    3.84    .768. 

10  Anstvers:  1  36.72  3.672  4.08.  2  2.38  357  23.8. 
3  39.67    3967    396.7.      4  9.6  ft.      5  10  panes.      6  $3.00,  cost  of 

I  bbl.    $9.75.      7  $.20    $2.95.      8  $1522^.      9  5  da.     10  20  bbl. 

II  2000  pencils  166|  doz.  12  3780  whites  294  colored  126 
Chinese.  13  174  sheep.  14  495  doz.  15  $21.21f  16  643ibu. 
21|  loads  $129.95^  profit.  17  2  yr.  2  mo.  7  da.  1  yr.  9  mo. 
15  da.       18  $74.66|.       19  $.95.       20  $738.       21  8.526  T. 

11  Examples  of  the  terms  odd  number,  even  mimber,  factor, 
prime  factor,  tmdtiple,  and  least  common  multiple,  will  have  to  be 
given  before  the  exercises  involving  those  terms  are  given.  A  good 
method  is  to  write  upon  the  board  examples  of  what  is  desired  to 
be  taught,  and  then  to  ask  the  pupils  to  give  other  examples.  After 
the  pupils  get  a  clear  idea  of  the  terms  they  may  be  asked  to  define 
them  in  their  own  words.  The  following  definitions  will  suggest 
to  teachers  the  kind  of  illustrations  and  questions  that  may  be 
used.  An  even  number  is  a  number  that  is  exactly  divisible  by  2. 
An  odd  number  is  a  number  that  is  not  exactly  divisible  by  2.  A 
factor  of  a  number  is  its  divisor.  A  prime  factor  is  a  divisor  that 
is  a  prime  number.  A  multiple  of  a  number  is  any  number  which 
it  will  exactly  divide.  A  common  multiple  of  two  or  more  numbers 
is  any  number  which  each  of  them  will  exactly  divide.  The  least 
common  multiple  of  two  or  more  numbers  is  the  least  number 
which  each  of  them  will  exactly  divide. 

12  The  exercises  of  this  page  should  be  practiced  upon  until  the 
pupils  can  give  the  required  answers  readily.  The  composite 
factors  may  be  given  first  if  necessary. 

13  In  teaching  how  to  find  the  prime  factors  of  large  numbers, 
use  small  immbers  first.  Two  ways  of  factoring  may  be  used,  — 
first  to  get  the  composite  factors,  and  then  the  prime  factors  of 


2 
2 
2 
2 
3 

240 

120 

60 

30 

15 

V.  14]  teachers'  manual.  85 

these  ;  thus  the  prime  factors  of  240  may  be  found  by  finding  the 
prime  factors  of  24  and  10,  or  12  and  20.  Another  way  is  to  divide 
the  feumber  successively  by  prime  numbers  thus : 

The  prime  factors  of  240  are  2,  2,  2,  2,  3,  and  5.  Lead 
the  pupils  by  examples  to  see  that  the  common  divisor  of 
two  or  more  numbers  exactly  divides  each  of  the  numbers  ; 
thus,  in  finding  the  greatest  common  divisor  of  8  and  12 
the  question  should  be  asked  :  "  What  number  will  exactly 
divide  8  and  12  ?  Is  there  any  other  common  divisor  ? 
Which  is  the  greatest  common  divisor  ?  "  Other  examples  should 
be  given  of  the  same  kind,  the  numbers  increasing  in  size  as  the 
lesson  proceeds.  The  kind  and  order  of  exercises  here  given  will 
suggest  to  teachers  what  should  be  done  before  each  new  set  of 
exercises  is  given.  Let  the  work  proceed  slowly  at  this  point,  and 
review  frequently. 

14—15  The  thorough  knowledge  of  common  fractions  to 
twelfths,  which  pupils  have  acquired  by  objective  teaching  and 
much  practice,  will  help  them  to  use,  intelligently,  fractions  of 
other  denominations.  By  means  of  the  cut  at  the  head  of  the 
page,  pupils  may  learn  something  of  the  relative  size  of  fractions 
of  various  denominations.  Lead  them  to  express  in  fractional 
form  any  given  part  of  the  unit,  indicating  the  fractional  unit  and 
the  number  of  such  units.  By  illustrations  and  comparisons  such 
as  are  here  shown,  lead  them  to  express  a  fraction  by  smaller 
terms.  Lead  them  to  see  and  to  say  that  in  such  a  change  of 
expression,  the  size  of  the  parts  has  increased  while  the  number 
of  parts  has  decreased.  In  the  same  way  teach  the  expression  of 
a  fraction  by  larger  terms,  and  why  the  value  is  not  changed. 
Drill  upon  exercises  similar  to  17,  page  15,  should  be  given  until 
the  pupils  can  tell  at  sight  to  what  common  denominator  any  two 
or  more  fractions,  whose  denominators  are  less  than  20,  may  be 
reduced. 

16—18  Frequently  the  common  denominator  of  such  fractions 
as  are  given  in  1  and  2,  page  16,  can  be  named  at  sight.  If,  how- 
ever, the  denominators  are  large,  let  their  least  common  multiple 


86  GRADED    ARITHMETIC.  [V.  19 

be  found  by  finding  all  the  different  prime  factors,  as  shown  in  the 
following  illustrative  solutions  : 

What  is  the  least  common  multiple  of  80,  72,  and  60  ? 
80  =  2X2X2X2X5  By  this  method  the  L.  C.  M.  is  the 

72  =  ^X^X^X3X3  largest   number   multiplied   by   the 

60  =  ^X^X$X^  factors    of   the    other   numbers    not 

80  X  3  X  3  =  720  =  L.  C.  M.     found  in  the  largest  number. 


80 

72 

60 

40 

36 

30 

20 

18 

15 

10 

9 

15 

10 

3 

5 

By  this  method  the  numbers  are  divided  by 
any  prime  number  that  will  divide  two  of  them 

Q  Tn 7\ — Tk         without  a  remainder.      The  divisors,  remain- 

^  -T7. 7, ^         mg  quotients,  and  undivided  numbers  are  the 

— —         factors  of  the  least  common  multiple. 

2X2X2X3X5X2X3  =  720. 

Constant  drill  upon  the  oral  exercises  given  on  these  three  pages 
should  be  given.  A  good  method  of  drill  is  to  let  the  pupils  repeat 
the  steps  orally;  thus,  in  24,  page  16:  "These  fractions  can  be 
reduced  to  twelfths.  -^^  and  -j^  are  \^,  and  i|  are  f  ^,  and  ^%  are 
f|-  =  2y\  or  2^."  By  degrees  the  pupils  may  be  led  to  state  only 
results;  thus  in  47,  page  16  :  ''§§,  ^1,  ||,  -^^Y  =  2§§  or  2|";  or 
even  shorter  than  this  :  "  30,  38,  68,  -^^%P-  =  2|."  Eeduction  to  sim- 
pler forms  and  to  mixed  numbers  may  sometimes  be  made  in  the 
midst  of  the  work  ;  thus,  in  7,  page  18  :  "f  +  ^-  less  |  =  f."  ^'-2  5 
+  f^^lg-fj  less  ^§  =  1^"^  or  Ij."  Short  exercises  of  this  sort  in 
which  every  pupil  is  actively  thinking  should  be  given  frequently. 

19  A7isivers :  1  2^.  2  4f.  3  2i^.  4  2^.  5  5§g.  6  2^^^. 
7  Hf  8  Iflf.  9  U-  10  1^\.  11  lAV  12  2/^V 
13  Ifa-         14  55|fl3.         15  2135.         16  737V  17  82||^. 

18  51f  a.  19  383tW  20  300^^.  21  254^1^  22  193§|f. 
23  176f-f|.  24  55|f  25  484f  26  19}|f.  27  10^|. 
28  ^^^  If  U-  29  T%\  ^%\  13.  30  ^V  m  m-  31  t\^ 
^V^  Ul-  32  13||  243f  39|-^.  33  198|§  109|f  23^. 
34  ll^f     14/t-     7^.      35  32^     164^^     2l3|.      36  24^     45^ 


V.  20]  teachers'  manual.  87 

8^t.     37  50,«T    99||    193|.     38  21,^    G^Uh    ^hh     39  18^ 
ll§i     10.       40  14^f     IT-Ht    1^     2r>f|.       41  85tYo     6^  11^^ 

41     5  4^     .S.S7  9  10043  0131  43  43     1113  139         4f)3  1 

10'i  4  1 

Some  of  the  above  answers  can  be  obtained  without  the  aid  of 
figures.  Encourage  the  pupils  to  use  as  few  figures  as  possible  in 
the  solution  of  problems. 

The  following  forms  of  written  solutions  are  suggested : 

25  27 

1fi2,-\  29     _1_110_L1244_12  3   3  86   998     949 

J-"3    f  T4¥     >     Tl4   T^  T44      I     T44   T44  ^T44  J'2 

Sjf  l  +  7  +  2fa=:10f|. 

20  Ansu-ers:  1  30}f|.     2  SStV^.     3  8/^.     4  5|f     5  lOSif. 

6     1  ■'5  7     17  a     2,^5  Q     5919  10    .Sli  11     15    643 

12  14^4/^.     13  If'o-      14  31//^.      15  241.      ig  71^3.      17  351. 
18  85i-i||.        19  28tV        20  57Ji-        21  39ff.        22  224i|i. 

23     1     27  04     18    59  01^     107  PR     9«    11  07    2  9         1       5  41 

254  67  83  3  55  969  OQ     61         11  2  3         147  29         ll 

273^        T5^        T4(y         TT7?         T573-  '^O     7^        ^^         24         -^?T?         40         •'^5  5 

1_2  5  1     5  9_  OQ    4.9  IS  2  3  fil9         .Q1370  861  4   3J1_        fil3 

8    7  2  30     883  9129         183  4         10811  1147  18    29  9fi725 

^T5T-  •'^    "2T(r       ^-'^Sl       -'-''35       •'■'^1547        J--'^T20         -'-''12^       -'"TOOH 

102  3  59  31     '>S    6  7           304  7         987  9         ^9    5  0  5           4r)16  9         39   9           Ql     5_ 

9118  29  32     fiOlS       802  5       482  4         A8    7  86           58   9           48709           58101 

80  2  3  0  5  33     15              13           317            147           II           171             137          14  5  3 

34     415  18487         A31         4205         811         462          fi43          8    72                   31^     8247 

91151  18  563         9076           11  5         1  8 1  1  •'>         2789           104321                 3fi     2847 

30102  1  9813  1        n3    2  5         4f;   9          39  2  5         31     5  9          99    54  5                37     (50    5  1 

802  6  3      48  7  1        fiO  3 1        582.      40  17       501  •'5  4      81   53  7  3«  4205 

189       ^61      4751        87      5  61        71        04         .3Q   822      2125     lOiii 

20    5  2         12      10    29  97   G  7  20  IT  7  40     984  3        3010        903  3        5318  7 

4fil7         33   7_        312  9  99    9  1  41     104  4  SOl^l         483  7  9  fiO    277 

585         40123  5       50    5  9  81  123  1  40     8fi3  1  04262         8.3103         70142 

00^       *<^54B¥      '^"^102^      "^-"-^TT^^?-  *^    '^"45        '^^■M5       "'^7"50        '  ^3TS' 

fi,3i  1      70i£o      873335      00  4  43   19215      30  73       24271     2.320 


88  GRADED    ARITHMETIC.  [V.  21 

191     22Hf     32t«A     27§H-       **  32//^    48i§|    34«H  56^11^ 

54^     36-rV     36H     SO^W-       45  73/JV    It^^uVu     54i|^  72^11^ 

66f    52fHi     64H     89^3^V       46  36if|     Sliff    47i|f  Tli^* 

57^    50^    57/5^:j    41HM-      47  77//^    llOifi    66|02  87if§^ 

69i     66|f J     85^5^?     lOOHfi-        48  97^3^7^     119HI  76|f|i 

1201195         10413         S0209         SQ31        1021877  4Q     111221  194.323 

111    3  1  1.S03  8  7  lOQll         10*^1         117455  190287  t^f)     15523 

1,S3_9-3J7  1Q049  19        14A6  19         1  91     5         US   429  1  15     8  9  _        170205 

-'-'-''^504(y        •^'^'-'T¥20        -"-^"sOS        -^"^l^       -'-^"1120        "'^^'^^228         -'-*^2024- 

51     _3  11         rin   flTT?  188  53  31  92  527  I^O       3  5  8 

**■*■      16"         45^        •'^^   '*'^''-         2  73        T2(T         7  0         TST  T5  7  3-  *''^     T44         'g'S 

181  8  1         131  7_  5_5__9  9_  Ik'i     31        i        14         4GX6         J7J!_        19.1 

B^B^O         B^5        B"       308         2^7         T25?4*  *'*^     72         8         40         5733  120         220 

141        _8_305  54     ,SG5       17297         529        93A1        711        ,^235       5lO         7140 

T^tf        10B45*         ^'^     '-'95        -^'^30         "^30       -^357         *T^       "^364       ^2T         *l43- 

55     3125       1731        51G7        9   887  74        Q63  5l        7  5fi     7209        904  1 

183j6_        1Q_30_59  1111         IS    11  9(^269_        101271  57     729         904 

-^"105         -'■•^1082^        -'--'-80         -'-"120         -^"1008        •'-^3S72-  ^'      '  5i        ''^^ 

18-a3_       1«    500         10^       17    59  95    67  5  iqi29  5Q    1  8  9_  Q     7 

12^12.  151292  Q2754  14-7  90    1  50     273  1  90    6  7 

9Q3  8  3         51137         4.59  81102         80    13  fiH     97    8  9  90    6  1  9Q      7 

^'^7  2  0         ^-"^S-^?         ^^T4         ^-"-TT^        '-*^T0  8-  "^     -*T44         ^'^T^H        -^24  0 

512093         1510         8119         2931         20123  9  fil      24318  1153         99407 

•^-•-6174         *'^2T        '^-'-44         ^^^V        '^'^T^ge-  ^•*-     ^^T603        •'■-'-55         —'720 

18    8  53  8SII34  2Sll  243  18   9  5  fiO       20   3  5  816  7  1 

^^1^^^  '->°3577T  ^"^5^  ^^^4  ^'-'BBS-  "^       ^'-'432  "T782 

10-9  8*^1017  1  8417  14-8  5  449  2184  2  5  63      fi42  2  1 

■^^5^^  '^-'TT6B2  "-"^^T  -'^^4TJ5  ^T44  ^e5'g2i-  "'*      "^2  24 

717^^   42^15   64iafi   50^^   4411^   52//^.   64  60}|| 

fi742  5  2917  9  18163  10  Ifi  1  8  5  80263  8*^3  17  (>119  503 

"'452  '^•^3  85  *"T'9T59  ^^222  ^'^6  7  2  ^*'244K  "•'-8905e- 

fi5      77118  792  56  8         f;83  7  63         50   46  2  5111  51     7  9  fiO    5  2  3  3 

66       IfiiOl  4.1549  842  23  1  10209  5_1_  21 -2_9_9_  28^*  2  1_7_ 

00  J^"l05  *TB?0  ^^7  92^  -'-'^SOS  ^12  '^-'-1120  -^"^546  84 

1  716  8  1  9 
-^'2  2  2^4" 

21    Answers:   1    260/5   ^-         2    31o4ia  bu.         3  583^5^   A. 

4  $.75.         5  $416911.         6   $6262|.         7  ^143.  8   f26if. 

9  43||  T.       10  $439.       11  21^9^  ft.       12  557if  ft.  13  $j%%. 
14  ^\. 

Let  the  pupils  perform  orally  such  problems  as  they  can  perform 
in  that  way. 

23  Ansivers:  1  |^.  2  194iJ  lb.  3  if.  4  39?  bu.  5  8.,Vmi. 
6  5  h.  15  min.  7  32f  a°.  8  30^  yd.  9  2J^  yd-  10  f  8,^- 
11  6I815  mi.       12  $15f.       13  o77|i.       14  69863^5  gr. 


V.  23]  teachers'  manual.  89 

Analyses  both  oral  and  written  should  be  required,  the  steps  to 
be  clearly  indicated,  and  what  each  result  stands  for.  The  method 
of  solution  in  some  cases  might  be  indicated  in  one  place  and  the 
work  in  another,  as,  for  example,  4  : 

16|  2  0 

251-  2  7 

100   bu.-(16|    bu.  +  25f  bu.         ^^^^      ^^ 

+  4011  l^u.-127  pk.)==amt.  of        ^^,  =  1^=91^ 
corn  to  fill  the  bin.  ^  ^^^        '^^      ^^J^ 

100  —  60^  =  39|  bu.  60^  bu. 

3,3  Let  the  pupils  practice  upon  this  illustrative  work  until 
they  can  clearly  see  and  state  the  two  ways  of  multiplying  a  frac- 
tion by  a  whole  number. 

24  Some  of  the  work  here  called  for  is  review,  but  it  is  advised 
that  the  objective  work  be  continued  until  the  pupils  have  a  clear 
idea  of  the  method  of  finding  the  fractional  part  of  any  number. 
After  the  lines  have  been  used  in  the  solution  of  problems,  the 
same  solutions  should  be  reviewed  without  objects  in  such  state- 
ments as  the  following  (14) :  "  ^  of  f  =  ^3^  or  ^ ;  s  of  |  =  5  times 
^  =  fl."  This  could  be  followed  by  rapid  silent  solutions,  the 
answers  only  being  given. 

35  Dividing  by  a  whole  number  is  only  another  form  of  ex- 
pression for  getting  the  fractional  part  of  a  number.  |-  of  f  or 
J  -^  2  is  obtained  in  two  ways,  as  shown.  Let  the  pupils  practice 
upon  this  with  illustrations  until  they  can  tell  readily  which  pro- 
cess to  use. 

36  —  37  Division  by  a  fraction  may  also  be  taught  by  the  aid 
of  disks  or  by  the  drawing  of  lines  or  squares.  For  suggestive 
illustrative  teaching  see  page  66  of  the  Manual.  The  same  kind 
of  illustrations  may  be  used  for  division  in  other  fractional  num- 
bers to  show,  first,  that  the  number  divided  and  the  divisor  are 
first  subdivided  into  parts  of  the  same  size,  or  reduced  to  the  same 
denomination.  For  a  time  this  method  of  division  may  be  employed. 
Afterwards  the  pupils,  knowing  that  the  quotient  depends  upon 


90  GRADED    ARITHMETIC.  [V.  28 

the  size  of  the  divisor,  may  analyze  problems  as  follows  (19, 
page  27):  ''|^l=f;  f-^^  of  1  =  5  times  f  or  ^^^  ;  ^-^1  = 
■J  of  -2/  or  II  =  1^^4."  Such  analysis  should  not  be  given  to  the 
pupils,  but  made  by  them  in  answer  to  questions  from  the  teacher. 
After  a  time  the  pupils  will  see  for  themselves  that  the  quotient 
obtained  by  dividing  by  any  fraction  is  the  same  as  the  product 
obtained  by  multiplying  by  the  same  fraction  inverted.  There 
can  be  no  harm  in  such  a  method  so  long  as  the  pupils  understand 
the  process.  In  a  final  review  of  these  pages  let  the  pupils  say, 
in  such  exercises  as  12,  page  27:  "-^  of  4f  =-J  of  -3_2  =  i^"  etc. 
And  in  such  exercises  as  15,  page  27:  "7  -^  §  =  7  X  ^  =  %^  =  4i." 
The  processes  of  the  solution  may  be  made  silently  and  only  the 
answers  given,  thus  :  "5-i--^  =  15;  i^i=: 2." 

38  —  30  Let  the  analysis  of  these  problems  be  simply  expressed, 
and,  as  nearly  as  clearness  will  permit,  in  the  pupils'  own  words. 
The  analysis  of  some  of  the  problems  may  have  to  be  preceded  by 
questions;  thus,  in  2,  page  29:  "One  third  of  a  yard  will  cost 
what  part  as  much  as  two  thirds  ?  If  you  know  the  price  of  one 
third  of  a  yard,  how  can  you  find  the  price  of  a  yard  ?  "  And 
in  15,  page  29:  "5  lb.  will  cost  how  many  times  as  much  as  2^  lb.? 
l;^lb.  will  cost  what  part  as  much  as  2^  lb.?" 

31  Answers:  1  8f  80^  1062  391f  656f.  2  81f  335a 
272|     1845     753f.       3  ll^V     20^     7U     100     12/^.       4  7i     8 

23i        45        .5-7-  5     _2JL  217  74  13  117  fil7  6  1 

^Og        to        UjQ.  i'     100         T8  0  0  2  85         3^0         5^00-  "     TSIJ        ^K         ^ 

iU  f  7  24||  67/3V  1791  162J|^  164||.  8  232^  261| 
292f  4101 1  123457^.  9  28 j^  455f  1943^  705f  426|. 
10  134iV  2542|  1945f  2604^  3539i72-  H  1-19^  398^  410f 
120f  33|g.  12  847|f  420§5  735023  2094/^  773^1^. 
13  187i  733|  888 «  242f  96^.  14  281^^^  273^  2356 
IS^f  85^.  15  152  253|  151f  409j3  458^^.  16  128^ 
24193-  1775/7  13545  1093^.  17  ISOif  1364^^  444|2 
1009,V  1055^.  18  inn  IS^^'V  1311  iejH  41'^  108\U 
40,Vo-  19  13611  444f  246?^  523/^^5  869|a  89j|§. 
20  2«3  10^^,  1323  40/^  2931^.   21  90  1055»g  3151^  687f 


V.  32]  teachers'  manual.  91 

82^5^.        22  88^^^-     188,5f}      13f     14511^        23  05,?     814     2871 
8202«.        24  73^^     OO^^V     25^     I573V      25  35 ^/v'  63 J^    93e 

11717  1 

33    Answers :    1  $44G8f .       2  345  qt.       3  2602|  ft.     8G7f  yd. 

4  $17.12f|.         5  $508800.         6  |G4280|,  wife  $24105^,  son 

$8035t-V,  hospital.          7  $18.28^.          8  )ii8.202V  9  fl057i. 

10  $1197^^.             11  -IFGOll.             12  |3.06i.  13  f313±|. 

14  $14.87|.           15  |29.56i.           16  $57.49i\.  17  $83.43. 

18  $5488GiV             19    ^'^Ul             20   $446i.  21   $8G7. 

22  $75.G3iJ.          23  $78.72a.         24  $15341f.  25  ^^^^      ^V 

Qi  ,S5  1  ^_1_  2fi       2^5  4  1_  .3  7  4  J.  07     14.1  G  3 

12  7  2_2J)^  5   _  1  OQ        _8  3 11055  L  13  21 

■55^5"  T080  15^!»4"4  T2?*  "*'     T50000  -^  T  5  U  ff  TiJ2(J  7400 

2§a||.   29  280  408   8f  47f   ^V^o-   30 .1G|-  20   IGei 
40tV  873§.    31  40f  58f  158if"  10|f   24i-f.    32  2|| 

1145  1  5Z         JL5_        143 

-^24  7         4  7  5  12  2         -^8f " 

Let  the  jmpils  be  accustomed  to  multiply  by  a  multiplier  placed 
in  any  position  ;  thus,  in  1,  the  solution  is  written : 

G|-     Cost  of  1  ton. 
650  $G|,  cost  of  1  ton,  X  G50  =  $3900 

3900  or,    +  $5G8f  =  $44G8f,  cost  of  G50 

5G8f  tons. 

$44G8f     Cost  of  650  tons. 

In  these  solutions  the  logical  multiplier  is  650,  but  the  number 
actually  used  is  6^.  The  multiplication  should  be  made  with  the 
multiplier  in  either  position. 

Cancellation  of  common  factors  in  both  dividend  and  divisor 
may  be  made  when  convenient ;  thus,  in  25  : 

^0 


6f  ^  400  : 

3  X0i^      ^"' 
20 

33    Ans7vers:    1  8|     155a 

5U|     %%     7||. 

2  8||    em 

Hu   6-tf   un-     3  tV? 

ll"i§     4fi|     7^1^ 

16ff.       4  23^ 

$.16^^^^    l.lSfH    •03§§|    $.30|£^.      5  ^\    f    2/^ 

.      6  U\t    6f 

92  GRADED   ARITHMETIC.  [V.  34 

S1015  7J3  71  115  ft1l3         129  69  0_  Q     5Q5_5  1  0.S 

#iUy^.  /   ^'oTT        'T?       -"-TF-       "    ^T4       -'■^B^       TOOT-         ^    •-'^'^^T       -"-^O 

<t)yY  t^*-*  J^^TO-  •*■"     300  ^T^5  '^T32  -^^147  -^^T04* 

11     QII6  I.SS22  .Si  8  5  Qfil  Q117  12     18         1-3  11  59 

1Q1191  49  2  47  14       588  212  QQi         1    46  T^     9 1  1 

565^  l;54ff.  16  lofii  lejif  2686f.  17  G2.S|-  457Ty5 
2||f.  18  450.  19  97^.  20  143^.  21  1756t;1i.  22  4ifia 
23  275. 

34  ^nsu-ers ;  1  125.  2  H-  3  •'i'2.40.  4  22^  bu. 
5  $.76f .  6  27.  7  $1.43  15f  lb.  8  20  bu.  $12,3^. 
9  25 /j  bu.  apples  60  bu.  10  222f  10  suits  4i|  yd. 
11  10^^  T.  $30f.  12  $5.62|  200  francs.  13  6  breadths. 
8  breadths.  14  7  bu.  8  bu.  5f  bu.  15  $2  gain.  16  1/^ 
$112^.       17  283^  gab     166.04^.       18  $68f|     22|  da. 

The  usual  forms  of  oral  and  written  analysis  of  these  concrete 
problems  may  be  made,  and  afterwards  they  may  be  written  out 
"on  a  line."  As  the  pupil  proceeds  with  the  oral  analysis,  he 
writes  the  number  above  or  below  the  line ;  thus,  in  10  the  pupil 
may  be  taught  to  say :  "  My  answer  is  to  be  in  yards,  so  I  write 
100  yd.  as  the  number  to  be  wrought  upon.  It  will  take  J^  as 
many  yards  to  make  1  suit  as  it  will  to  make  18  suits.  I  write 
18  below  the  line  as  divisor.  It  takes  so  many  yards  to  make  1 
suit.  To  make  40  suits  it  will  take  40  times  as  many  yards  as  it 
takes  for  1  suit.  I  place  40  above  the  line."  The  problem  when 
finished  appears  thus  : 

20 
100  yd.  X  40  ^  2000 

X$  9     J    •      -^  -9- 

9 

35  Some  work  with  objects  may  help  the  pupils  to  learn  this 
new  principle  of  finding  the  whole  when  a  part  is  given.  The 
illustrative  work  given  will  show  wliat  should  be  done  with  the 
objects.  Give  the  pupils  a  certain  number  of  counters,  as  6,  and 
tell  them  to  show  you  -J  of  the  counters ;  f  of  them  ;  §  of  them. 
Then  give  to  each  pupil  4  counters,  and  say  that  they  have  f  as  many 


v.  36]  teachers'  manual.  93 

as  you  have  in  your  hand.  Ask  them  to  point  out  -^  as  many  as 
you  have,  and  then  ask  them  how  many  more  counters  they  must 
take  to  have  as  many  as  you.  Much  illustrative  work  with  dots 
or  marks  will  help  to  fix  the  principle.  To  be  sure  that  they 
understand  it,  give  them  work  of  both  kinds  —  a  part  given,  to  find 
the  whole,  and  the  whole  given,  to  find  a  part. 

Let  the  analysis,  even  of  those  problems  which  the  pupils  can 
perform  orally,  be  written  out  in  full.  It  will  be  seen  that  the 
conditions  vary  considerably  and  need  careful  thinking. 

3G  When  any  new  feature  or  condition  is  given,  as  in  9,  let  the 
pupils  try  the  problem  first  before  attempting  to  assist,  or  even 
before  calling  their  attention  to  the  new  condition.  If  they  say, 
as  they  will  be  likely  to  at  first,  that  20  gal.  is  f  of  the  whole 
number,  simply  ask  them  how  many  gallons  were  left  and  what 
part  was  left ;  and  so  lead  them  to  think  what  part  of  the  whole 
20  gal.  is  equal  to. 

37    A7iswers:   1  $.51.  2  20|  wk.  3  A,  $45       B,  $15. 

4  A,  $20  B,  $120.  5  .4  T.  .096  T.  6  32  bbl.  7  30  peaches. 
8  24  yd.  9  $17.  10  $21i.  11  $.06  $4.52.  12  $.52|. 
13  $.23^.  14  $8000.  15  $200.  16  $1000.  17  $333^. 
18  $47i|. 

The  pupils  should  be  asked  to  make  written  solutions  on  a  line 
with  oral  analysis,  or  to  make  written  analyses  with  each  step 
indicated  and  each  result  designated.  These  two  methods  are 
shown  in  the  following  solutions  of  11 : 

8)  $0.64    cost  of  2|  lb. 


.08 

i( 

"     ilb. 

3 

$0.24 

u 

"  1    lb. 

18| 

192 

24 

20 

4 

$ 
0^/  X  $  X  113 

t 


=  $4.52. 


^4.52    cost  of  185  lb. 


94  GRADED    ARITHMETIC.  [V.  38 

Encourage  the  pupils  to  perform  orally  as  many  of  tliese  and  the 
following  problems  as  they  can. 

38  Ansivers:  1  |  cd.  Ij  da.  2  2|  da.  2  da.  If  da.  3  $.30 
$.36.  4  $96.  5  $4.20.  6  $44000  $13750.  7  40  pupils. 
8  $48000     $36000,  widow.     9  $.34f.      10  $.69tV5.      11  $8.40. 

12  6  weeks.       13  157U  T. 

39  Answers:  1  7^^  h.  2  191^- A.  3  $.80.  4  $60480. 
5  $16853^      B,  $8426f.         6  154/^  lb.      $19.29.         7  113f  mi. 

8  3  da.     9  2  da.     10  \%  da.     12|f  da.        11  $750.      12  $4800. 

13  $140^.       14  76  shares     $14. 

40  Ansivers:  1  $753.81^.  2  $63x1^.  3  $17t'VV-  *  ^^H- 
5  $1579f.  6  $2176585.  7  $2508.  8  $86.62^.  9  House, 
$7058-^-  Land,  $4235i.  10  6§  T.  11  16  s.  (|  yd.  left). 
12  $25116|.  13  $15,073+.  14  |  lb.  15  .07  lb. 
16  $45.26.        17  27i|  jars.        18  f      j%. 

41  Answers:  1  f|.  2  $38.76+.  3  249|  A.  4  $48f. 
5  $13621.        6  66||  yd.        7  $4.41|.        8  37f|f  h.        9  $6.50. 

10  $5f  |.       11  42if  I  h.       12  162^  bu.       13  $468f.      14  3  yr. 

11  mo.+     2  yr.  4  mo.+        15  41if^.       16  $8^. 

43    Answers:   1    $306^1.         2    $11271- gain.         3    $82.50+. 

4  Po.,  500  mi. ;  Kan.,  900  mi.  ;  Yel.,  1000  mi. ;  Ked.,  1200  mi. ; 
Ark.,  2000  mi. ;    St.  L.,  2200  mi. ;    Nile,  3300  mi. ;    Am.,  3600  mi. 

5  1761b.     8f  lb.       6  20  yr.       7  100  ft.       8  32 j/:^  gal.     65^^  lb. 

9  To  Sar.,  180  mi. ;  to  R.,  360  mi. ;  to  B.,  480  mi. ;  to  A.,  720  mi. ; 
to  L.,  900  mi. ;  to  J.,  1500  mi.  ;  to  S.  L.  C,  1800  mi.  ;  to  S.  F., 
2250  mi. 

43  A  review  of  the  objective  work  in  decimals  to  thousandths, 
given  in  Book  IV.,  will  be  found  useful  in  giving  a  good  foundation 
for  the  work  here  given.  If  the  pupils  clearly  understand  the 
expression  and  use  of  decimals  to  thousandths,  there  need  be  no 
use  of  objects  in  dealing  with  higher  denominations.     Dwell  upon 

6  and  7  until  they  are  well  understood.  If  9  is  found  too  difficult 
it  may  be  omitted  for  the  present. 


V.  44] 


I'EACHERS     MANUAL. 


95 


14  19.9991. 

18    111.8909. 

.03     .64     1.33 


44  Lead  tlie  pupils  to  see  why  the  annexing  of  a  cipher  at  the 
right  of  decimals  does  not  change  their  value. 

45  Answers:  1  7576.13558G57.  2  !i?99.'.)99  $99.9995. 
3  210.962803.  4  3.37079  in.  5  60.8533^  rd.  6  59.0135  A. 
7  390.95347.  8  652.6939.  9  837.363.  10  1.0425  .1953. 
11  280.4274.  12  9.9191.  13  1005.991201. 
15  1.94.  16  114.194.  17  94.61359. 
19  72.7699.  20  4.2  .42  .042  7.2  1.62.  21 
.021     6.4.       22  .0101     .1836     .624     .576     .02108. 

Some  preliminary  exercises  may  be  given  to  show  that  hundredths 
of  hundredths  give  ten  thousandths,  and  that  thousandths  of  hun- 
dredths give  hundred  thousandths,  etc.  This  may  be  shown  by 
common  fractions. 

It  is  not  necessary  after  the  first  few  exercises  to  explain  each 
step  in  addition  and  subtraction.  Work  of  this  kind  in  one  part  of 
the  decimal  system  ought  to  be  no  more  difficult  than  similar  work 
in  any  other  part. 

46  Ansivers:  1  35.77.  2  .3431.  3  3.9786.  4  12.353536. 
5  14.404968.      6  2432.5.      7  84.78384.      8  26.3088.      9  8298.63. 


10 

74.2118. 

11    8.64838. 

12    49.0245.          13    81.2448. 

14 

793.074. 

15   3.10156. 

16   .0095823.         17    .24609102. 

18 

.976050. 

19   8476.4072. 

20    67.268.         21    22.25562. 

22 

9.322987. 

23    9247.854. 

24   164.467.         25  2370.104. 

26 

181.80208. 

27    6.0888. 

28    2470.713.            29    4.8. 

30 

1564.9236. 

31  .003267 

.93654        .0772101        2.228094. 

32  6.1509  .061509  6.09609.  33  .040068  .0941598  2.1005649 
.060102.  34  .0144684.  35  369.84414285.  36  .108  1.08 
.01188    .0108108   10.8.     37  100040.    100040.10004.     38  .6501755 


.0650390        .0078520        .262626 
40  803.8024.         41  3.5000056. 
44  10924.914.      45  1312.6476. 
48    684.39378.         49    61.2612. 
52  .203775.       53  .58313850. 

47    Ansivers:     1    88.787. 
4   878.17221.  5   296.478, 


.0135395. 
42  10000.84. 
46  44850.3102. 
50    160160. 
54  520.96. 
2    94.261544. 
6    525.5491. 


39  2050.3935. 

43  .087675. 

47  616.9472. 

51    .8757567. 

3    1424.08476. 
7    7852.547. 


96  GEADED   ARITHMETIC.  [V.  47 

8  214.9854.  9  7871.0072.  10  52413.821.  11  2551.6278. 
12  104066.5212.  13  633.8787.  14  1052.62201.  15  .1984512. 
16  32.1082179.  17  981.  18  7006.3.  19  109.46.  20  518.1357. 
21  488.08032.  22  3793.9408.  23  2593.44.  24  80012.6. 
25  1738.8  yd.  26  $8154.675.  27  1153.705  mi.  28  $112.4475. 
29  $1038.744.  30  $37.0476.  31  $34,832.  32  $2044.80. 
33  $3548.12.  34  $35574.59435.  35  .2  .02  .004  .4  3.4 
.00004  3.334  .00034  .0000004.  36  20  200  20000  .2  .02 
300000  .003  3000000  3000.  37  .006  .6  10  .01  .001 
.0001     1     .1     9.6. 

Before  the  exercises  in  division  of  decimals  are  begun,  there 
should  be  much  practice  upon  such  work  as  the  following  :  "  Tenths 
of  hundredths  give  what  ?  hundredths  of  tenths  ?  hundredths  of 
hundredths  ?  tenths  of  thousandths  ?  thousandths  of  hundredths  ? 
thousandths  of  thousandths  ?  tenths  of  tens  ?  hundredths  of  tens  ? 
tenths  of  hundreds  ?  hundredths  of  hundreds  ?  etc.  Tenths 
divided  by  tenths  give  what  ?  hundredths  by  hundredths  ?  units 
by  tenths  ?  tenths  by  hundredths  ?  hundredths  by  tentlis  ? 
thousandths  by  hundredths  ?    thousandths  by  tenths  ?  "  etc. 

In  division  of  decimals  there  are  three  possible  cases,  viz.  : 
(1)  Division  in  which  the  divisor  and  dividend  are  of  the  same 
denomination  ;  e.g.  .6  -r-  .3.  (2)  Division  by  a  decimal  in  which 
the  divisor  is  of  a  lower  denomination  than  the  dividend  ;  e.g. 
.8  -f-  .2  ;  .8  -f-  .04.  (3)  Division  by  a  decimal  in  which  the  divisor 
is  of  a  higher  denomination  than  the  dividend ;  e.g.  .04  -^-  .2 ; 
.008  -i-  .04,  The  first  of  these  cases  ought  to  give  no  trouble  to 
the  pupils.  It  is  readily  seen  that  4  tenths  will  be  found  as  many 
times  in  8  tenths  as  4  quarts  in  8  quarts,  etc.  That  is,  they  see 
that  when  the  divisor  and  dividend  are  of  the  same  denomination 
the  quotient  is  a  whole  number  (of  times).  Neither  will  there  be 
difficulty  in  dividing  by  a  whole  number.  The  pupils  see  at  once 
that  .08  -^  4  is  :J-  of  8  hundredtlis  =  2  hundredths,  which  is  the 
same  denomination  as  the  dividend.  All  other  classes  of  problems 
in  division  may  be  easily  resolved  into  one  or  the  other  of  the 
classes  already  understood.     For  example,  if  it  is  required  to  find 


V.  48]  teachers'  islanual.  97 

how  many  times  4  lumdredtlis  is  contained  in  2  tenths,  the  operation 
would  be  expressed,  .2  -~  .04,  or  .04)  .2.  The  pupils  know  that  a 
cipher  or  ciphers  placed  at  the  right  of  decimals  do  not  change 

their  value,  and   therefore    express   the  operation  thus :    "     ■^        . 

5 

All  such  problems  may  be  performed  in  a  similar  way.     Again,  if 

it  is  required  to  find  what  part  of  .2  is  .04,  the  operation  would  be 

expressed,  .04  -i-  .2,  or  .2)  .04.     The  pupils  should  know,  that  when 

the  divisor  and  dividend  are  multiplied  or  divided   by  the  same 

number,  the  quotient  is  the  same.     Therefore,  by  multiplying  the 

divisor  and  dividend  in  this  problem  by  ten,  the  operation  may  be 

x2.)  X  0.4 
expressed  thus  :       ''^ '—  •    Really,  the  pupils  should  be  sufficiently 

familiar  with  multiplication  in  the  various  denominations  to  know 
that,  when  the  dividend  is  hundredths  and  the  divisor  is  tenths  the 
quotient  must  be  tenths,  or  when  the  dividend  is  thousandths  and 
the  divisor  is  tenths  the  quotient  must  be  hundredths,  etc. 

After  several  problems  have  been  done  in  this  way  the  pupils 
will  see  that  they  can  point  off  as  many  decimal  places  in  the 
quotient  as  the  number  of  decimal  places  in  the  dividend  exceeds 
the  number  of  decimal  places  in  the  divisor. 

48    Answers :    1  a  .8002  ;     I  32000  ;      c  834400  ;     rf  3  ;     e  .9. 

2  a  3000.375  ;     h  1066.666+  ;     c  .0754+  ;     d  .00301 ;     e  .00008. 

3  a  1066.666+  ;        h  26900  ;        c  6680  ;        d  255  ;        e  .0433+. 

4  a  116.5263  ;         h  47.946  ;  c  2.08  ;        d  .2726+  ;       e  .084. 

5  a  1.3333+ ;       i  16000  ;      c  1600  ;      cZ  82000  ;       c  146.3902  + . 

6  a  9050;         Z.  960  ;         c  .0463  ;  fZ  9.26  ;  e  1027.7279+. 

7  a  .8  ;  Z>  .08  ;  c  80  ;  rZ  800  ;  e  8000.  8  a  1402.5  ; 
h  140.25  ;  c  140.25  ;  d  7.0125  ;  e  140.25.  9  a  2090  ;  Z.  .02  ; 
c  2000 ;  d  4.06 ;  e  .5236.  10  402.3739.  11  200170. 
12  48.63794+.  13  .0092.  14  2980.  15  a  1.4634+; 
h  .0047+;  c  499.5004+ ;  d  .7976;  e  14.1843;  /  50000  ; 
^133.3333  +  .  16  a  54.225+;  i  13.8910+;  c  548.5301  ; 
(Z  196176.470+ ;  e  496761.904+ ;  /  30.2641 ;  ^333.3333+. 
17  a  4073;  h  27900+ ;  c  17.2558+;  d  90.065;  e  9008; 
/  344.5783.      g  .1568+. 


98  GKADED   ARITHMETIC.  [V.  49 

49  Answers:  1  a  .4673;  h  2.0925;  c  .00007+ 
d  502588.94+  ;  e  8.014.  2  a  .0000686+  ;  b  59.0277  + 
c  49.975+  ;  d  1795.537+  ;  e  8612.8345  +  .  3  a  2.2168  + 
b  63.8297+  ;  c  .5602+  ;  d  .00541+  ;  e  4664.067+.  4  100  yd 
1000  yd.  110  yd.  5  12000  apples.  6  20  yd.  200  yd.  250  yd 
7  12  qt.     19f  qt.     32  qt.     2000  qt.           8  7  da.     92  da.     800  da 

9  $4.30.       10  $4.20.       11  8.2+  h.        12  44.6  mi.       13  650  A 

14  90  pr.     800  pr.  15  240  coats.  16  23  da.     605.7+  da, 
17  25  cd.         18  $1.50      $150.75. 

50  Ariswers:  1  120  casks.  2  50900  yd.  3  5.84. 
4  504.6  boxes.  5  42.5  A.  6  87.9  A.  7  .5  .75  .125  .625 
.875  .375.  8  .05  .075  .08  .0625  .075  .15.  9  .15  .16 
.38095+  .1875  .9375  .4218  +  .  10  1.4  1.16|  1.8  2.025 
1.5625.  11  1.75  2.875  14.7619+  21.9375  40.015. 
12    15.3125     20.4     15.53125     27.1875.         13    i     f^     ^k     tI? 

7  1  1Q1  14.143  _547_        4_2_43_         _2_5_5J[_  15    304-2-^-3- 

¥0         ?TJO         -'^•^^-  -^^     ^^T~S^         50000         ^4  000  2500  0*  ■^*'    ""^5  0  0  0 

^44  57    11  1      100  1        16  _!_    _1_    1_3„    ^9_ J _401_ 

"^500^    -'-^400    20000-       •*■"  80    5O0    "^4  0    ^00    2000    50000" 

17     11  12  1  1_  18 T 323         ^3_        _^3Ji.        11. 

■*•'     ¥        ■J^         T5         ^        ¥00         TSO-  ■*■"       8000         ^30         ?00         400         "^4 

T-1^.  19  2.35.  20  7.918.  21  10.5785.  22  $1,728. 

23  $8f     $110.83+.       24  14.6  yd. 

In  clianging  common  fractions  to  decimals  lead  the  pupils  to 
think  of  the  fraction  as  another  form  of  expression  for  division ; 
thus,  f  =  3  -^  4  or  i  of  3. 

51  Ansivers:  19.6875  yd.  2  80  sq.  rd.  40  sq.  rd.  60  sq.  rd. 
100  sq.  rd.  3  38  A.  94  sq.  rd.  18  sq.  yd.  1  sq.  ft.  50.4  sq.  in.  9  A. 
128  sq.  rd.  150  A.  112  sq.  rd.  4  A.  126  sq.  rd.  12  sq.  yd.  -^%  sq.  ft. 
7  sq.  yd.  6  sq.  ft.  100.2  sq.  in.  4  2.0225.  5  52.545  mi.  6  $565.88 
$126.     7  $263.4375.     8  67  lb.  1  oz.  12  pwt.  12  gr.     9  5017.95  lb. 

10  15.         11  2.803+  oz.        12  i      I      ^*,      -^S      ^\%.        13  .3 
.142857      .16      .09      .2916.         14    l^V      tjV      tWit      itV      l^'^- 

15  yV     :jV     555 7j- 

In  teaching  repetends  let  the  pupils  see  by  examples  that  the 
figures  of  a  repetend  express  the  numerator  of  a  common  fraction 
having  as  many  nines  in  the  denominator  as  there  are  figures  in 
the  repetend;  thus,  .16  =  .If,  and  .108  =  Jgf. 


V.  52] 


TEACHERS     MANUAL. 


99 


53    Answers 


1 

2 
3 

5 

6 

7 

S 

9 

10 

11 

12 

13 

U 
15 
16 
17 
18 
19 
20 


460 
80 

708 

80 


450 

70 

800 

1004 

350000 

20 

48 

700 


2 

3 

.08 

.75 

.0075 

.60()6 

.03006 

.375 

.00875 

.3125 

.000076 

.5555 

.0098 

.5833 

.0000708 

.7333 

.000604 

.3895 

.190 

.7083 

.000087 

.6333 

.10050 

.425 

.070 

.18 

.545 

.3166 

.07 

.2 

.009 

.0583 

.4 

.0622 

.00070 

.0045 

.000010 

.127 

.070 

.059 

.50 

.001 

^TTTJ 

3 

ZTiV 

7 

^TJTT 

33 

¥(T 

mu 

85^5 

39 

■5T)(T 

2  1 

■J0I)TJ 

48/^ 

8  1  » 

1 

T25Ty 

141 

2(J^ 

800f 

70Tk 

900211 

so 

'^^hho 

1   3 

-'sooo 

^i. 

3  fi  0  3 

ST)Ty(T 

1  1 

4.115 
6.009 
180.035 
6.861 
20.2867 
88.083 
3.0789 
700.8105 
62.478 
8.15676 
3.0218 
1.707 
800.6075 

70.215 
930.67 
74.013 
1.2415 
8.064 
8.7507 
101.800 


.785 
.6726 
.41 
1.1375 
20.7615 
8.6233 
.8113 
.4 
48.7683 
8.7093 
.4258 
.885 
800.9106 

70.208 
900.6383 
70.0672 
1.0051 
8.167 

.7802 
1.009 


4.83 
6.6096 
180.375 
0.3485 
.6362 
80.6263 
3.7342 
701.1895 
15.1263 
.71409 
3.446 
1.182 
.3241 
.407 
30.1483 
4.0702 
.2454 
.151 
8.0897 
100.801 


1 

2 
3 

4 

5 
6 
/ 

8 
9 
10 
11 
12 
13 

U 
15 
16 
17 
18 
19 
20 


460. 
80, 

708. 

100, 

8, 


48, 
458, 

70, 

801, 

870, 

1904, 

350070, 

1, 

28, 

48, 

701, 


8 

115 

0135 

06506 

83375 

200076 

0498 

0780708 

011104 

25 

,076087 

1013 

775 

145 

078 

.589 

405 

0013 

04001 

7906 

508 


404 

80 

888 

6 

80 

80 

3 

700 

14 

450 

3 

71 

800 

1034 

350004 

20 

56 

801 


.16 
.0105 
.03006 
.04475 
080776 
.0528 
.0009708 
800604 
608 

080847 
1215 
.072 
.5525 
277 
099 
.408 
.2416 
.02401 
.1001 
.30 


10 

460.83 

80.6741 
708.40506 
.32125 
80.55576 
.5931 
.7333708 
.390104 
.8983 
450.633387 
.5255 
70.25 
.8616 
800.27 
1004.0073 
350000.4622 
.0052 
20.12701 
48.1290 
700.501 


11 

4.045 

5.997 
179.905 
.5.211 
20.12.53 
72.003 
2.9229 
700.7895 
33.642 
7.99524 
3.0202 
.297 
800.5925 

69.801 
870.49 
65.997 
.7597 
8.016 
7.3095 
99.792 


100 


GRADED   ARITHMETIC. 


[V.  62 


12 

1  460.045 

2  80.0015 

3  707.99.506 

4  .81625 

5  59.794076 

6  8.0302 

7  .0779292 

8  .009896 

9  47.87 

iO  441.924087 

11  .0997 

i^  69.365 

13  800.055 

i.^  730.062 

i5  103.429 

16  349930.395 

17  .9999 

18  11.96001 

19  47.3494 
^0  699.492 


13 


456 
74 

528 

6 

79 

80 

3 

700 

14 

449 

2 

69 


799 

973 

349996 

19 

40 

599 


0045 
.03006 
02725 
919376 
.0332 
0008292 
.799396 
.228 
919327 
.9205 
068 
5375 
863 
919 
392 
2402 
97601 
.0399 
.7 


14 

.35 

.06 
.35 

8.25 
202.06 
80.4 
.78 
.105 
480.6 
80.76 
.008 
7.05 
8006. 

700.08 
9005.8 
700.05 
10.006 
80.40 

7.206 
10.08 


15 

35. 

6. 

35. 

825. 

20206. 

8040. 

78. 

10.5 

48060. 

8076. 

.8 

705. 

800600. 

70008. 

900580. 

70005. 

1000.6 

8040. 

720.6 
1008. 


16 


73. 

108. 

3240. 

108. 

1. 

1440. 

54. 

12614. 

259. 

1. 

54, 

18. 

3 

541, 

72, 

4, 

144. 

1814, 


44 
054 

648 

4.526 

774 

0162 

4 

524 

45368 

378 

036 

135 

726 

62 

144 

3362 

432 

5418 

40 


1 

2 
2 

4 
5 
6 

7 
3 
9 
10 
11 
12 
13 

U 
15 
16 
17 
18 
19 
20 


17 

4.6008 
.800075 

7.0803006 
.0000875 
.80000076 
.000098 
.000000708 
.00000604 
.00190 

4.50000087 
.0010050 
.70070 
.00545 

8.0007 
10.04009 
3500.004 

.0000007 

.20000010 

.48070 

7.005 


18 

50.14872 

8.7208175 
77.17527654 
.00095375 
8.720008284 
.0010682 
.0000077172 
.000065836 
20.71 

49.050009483 
.0109545 
7.63763 
.059405 
87.20763 
109.436981 
38150.0436 

.00000763 
2.18000109 
5.23963 
76.354 


19 

.1428 
.036018 
6.3 

4.9797 
1.6306242 
643.54572 
.2340702 
7.3584 
692.92908 
.65221776 
.0024168 
.70641 
6.0045 
14.491656 
27098.4.522 
280.58004 
.24104454 
.19296 
5.78649006 
101.6064 


V.  52] 


TEACHERS  MANUAL. 


101 


3 

k 

5 

6 

7 

8 

9 

10 

11 

12 

13 

U 
15 
16 
17 
IS 
19 
20 


16, 

24! 

1616, 


9, 
3634 

49! 

436, 

5601 L 

904190, 

24501778, 


160, 
34 

706, 


20 

10280 

480045 

7810521 

00721875 

049535656 

078792 

0000055224 

000006342 

1314 

200702612 

0000804 

39935 

327 

30056 

46522 

002 

00070042 

8000804 

639242 

104 


21 

.0035 
.0006 
.0035 
.0825 

2.0206 
.804 
.0078 
.00105 

4.806 
.8076 
.00008 
.0705 
80.06 

7.0008 
90.058 

7.0005 
.10006 
.8040 
.07206 
.1008 


22 


23 


40.8 

4.6008 

60.03 

.800075 

1800. 

7.0803006 

60.36 

.0000875 

.807 

.80000076 

800.43 

.000098 

30.009 

.000000708 

7008. 

.00000604 

144.18 

.0019 

.8076 

4.50000087 

30.21 

.001005 

10.02 

.7007 

.075 

.00545 

2.07 

8.0007 

300.9 

10.04009 

40.08 

3500.004 

2.409 

.000007 

.24 

.2000001 

80.301 

.4807 

1008. 

7.005 

24 

1  460080. 

2  80007.5 

3  708030.06 

4  8.75 

5  80000.076 

6  9.8 

7  .0708 

8  .604 

9  190. 

10  450000.087 

11  100.5 

12  70070. 

13  545. 
U  800070. 

15  1004009. 

16  350000400. 

17  .7 

18  20000.01 

19  48070. 

20  700500. 


25 

136. 

200.1 
6000. 
201.2 
2.69 
2668.1 
100.03 
23360. 
480.6 

2.692 
100.7 
33.4 
.25 
6.9 
1003. 
133.6 
8.03 
.8 
267.67 
3360. 


26 

70. 

12. 

70. 

1650. 

40412. 

16080. 

1.56. 

21. 

96120. 

16152. 

1.6 

1410. 

1601200. 

140016. 

1801160. 

140010. 

2001.2 

16080. 

1441.2 

2016. 


27 


116.5714+ 
1000.5 
5142 

7 


9 

38, 

66742 


3776 
1 


11 

100 


8514  + 
3163  + 
0039  + 
95.55  + 
4730  + 
857+ 
3 

01 
25 

4212  + 
000009  + 
0029  + 
,0334  + 
0572  + 
2407+ 
0029  + 
1436+ 


102  GRADED    ARITHMETIC.  [V.  53 

53^w5fc'e?.s;  1   1449.0129.  2  5.049150.  3   143.285  A. 

4  9.85.  5  21i  yd.  91  yd.  84/^  yd.  20  yd.  6  17^  108-rV 
$7,368.  7  28.7  T.  8  $84.69+  $415,181  $16988.81. 
9  16.4008324.  10  $1135.3375.  11  33.4  da.  12  $30483.60 
$17419.20.  13  $46,426  $1,946.  14  65.78+ da.  15  $5404.20. 
16  26666.66+  sq.  ft.     120000  sq.  ft. 

54  A7istvers:   1  4.11375.      2  48.      3  $14,748+.      4  Is^^^  bbl. 

5  $7.3125.        6  $.0002     250000.        7  1466.542  bu.        8  2'l.825. 

9  $42.59025.  10  $5,405+.  11  $1.53.  12  357.88839. 
13  $27,133  +  .     14  697.     15  1366.326  bbl.     16  9|  mi.     3000  mi. 

55  Anstvers :  ^  l^bil\  it.     5  102  ft.     1224  in.     9  15  yd.  2  in. 

10  1  mi.  4720  ft.  11  2  mi.  269  rd.  5  yd.  12  ^V  t'o- 
19  3732  in.  20  2654  ft.  21  10989  ft.  22  1  mi."  280  rd. 
1|-  mi. 

The  pupils  are  supposed  to  have  had  some  practice  in  reduction 
before  beginning  this  section.  (See  Section  VI.,  Book  IV.)  Many 
of  these  problems  should  be  performed  orally,  but  the  solutions 
may  be  written  out  in  full  for  the  sake  of  learning  a  good  form  of 
written  work. 

56  Answers:  1  5f  mi.  2  264  paces  2112  paces.  3  30.6  mi. 
4    1485  paces.  5    198t\  rd.  6    1  mi.  208  rd.  2  yd.  2  ft. 


7  208t»9  times. 

8  15  rd.  2 

yd.  4  ft 

.        $5. 

87^. 

9  $1,947. 

10  $12.       11 

66  ft.     792  in. 

12  7. 

92  in. 

57    Answers 

:  .•    1    396  in. 

617.76 

in. 

2    5 

ch.         80  ch. 

3  3659.04  in. 

304.92  ft.        4 

498  ch. 

32868  ft. 

5  83952  in. 

6  70604.88  in. 

7  2  li.  4.16  in.     6  li. 

2.48  in, 

8 

17  li.  5.36  in. 

75  li.  6  in. 

9   2  ch.  40  li. 

6  ch. 

80  li. 

10 

16  ch.  84  li. 

11    1  ch.  7  li. 

2.56  in.         1  ch. 

26  li.  .( 

}8  in. 

12 

16  ch.  50  li. 

13    23  li.  7.84 

in.      2  ch.  27  li. 

2.16  in. 

14 

1  ch. 

15    12  ch. 

16  541.2  ft. 

17  1056  ft. 

18  2811.6  ft. 

53  mi.       3jJ'j  min. 

19  2f  min.     .35+  mi. 

Measurements  by  surveyor's  measure  are   not  generally  made 
now,  and  the  problems  here  given  may  be  omitted  if  thought  best. 


V.  58]  teachers'  manual.  103 

Exercises  with  the  surveyor's  measuring  tape  may  be  substituted. 
On  this  tape  are  given  the  feet,  and  tenths  and  hundredths  of  feet. 
,  Pupils  are  supposed  to  have  been  taught  angle,  triangle,  and 
rectangle,  and  to  have  had  some  practice  in  finding  areas.  (See 
pages  84-86,  Book  IV.)  Some  of  the  descriptions  here  called  for, 
however,  are  new,  and  the  figures  may  have  to  be  taught  again. 
Let  the  descriptions  be  made  by  the  pupils,  and  let  them  be  made 
from  the  objects  or  illustrations.  For  example,  the  rectangle  may 
be  shown,  and  attention  be  called  to  the  number  of  sides  of  the 
figure  and  to  the  kind  and  number  of  angles.  From  these  facts 
the  pupils  are  led  to  say  that  "  a  rectangle  is  a  four-sided  figure 
having  four  right  angles." 

5S  In  explaining  the  process  of  finding  the  area  of  a  rectangle 
the  pupils  should  be  led  to  multiply  the  number  of  square  units  in 
a  row  by  the  number  of  rows,  as  previously  shown.  (See  Manual, 
pp.  49,  75.) 

59  A7iswers:  1  10  A.  2  1  A.  140  sq.  rd.  3  3  A.  93^  sq.  rd. 
4  106f  sq.  rd.  32261  sq.  yd.  5  104544  sq.  ft.  6  585446.4  sq.  ft. 
13.44  A.  7  1424025.1872  sq.  ft.  32.69  A.  8  4  yd.  0  ft. 
9  45.375  ft.  10  21a  yd.  11  7^}  in.  12  f  762.30.  13  ^64. 
14  6422  sq.  yd.  112.L  yd.  241  sq.  ft.  15  21a  yd.  2§  rolls. 
16  ;$38.25+.       17  11  yd.       18  I846.28+. 

If  the  pupils  find  difficulty  with  exercises  in  which  the  area  and 
one  dimension  are  given  to  find  the  other,  lead  them  to  see  the 
process  from  simple  illustrations.  (See  page  83,  Book  IV.)  The 
drawing  of  plans  in  illustration  of  problems  should  be  required 
when  needed. 

60  Ansivers :  8  432  sq.  ft.  9  5400  sq.  ft.  10  6  rd. 
11  10  ft.  12  8712  ft.  13  4356  sq.  ft.  14  10.  15  4-  i  ^i^. 
16  ^%  25  i.o|.  17  .25  .01875  .058^.  18  .0075+ 
.1628+     .339+.       19  i     ^V     U-       20  .00625     .875.        21  1.8. 

The  terms  horizontal  and  vertical  as  applied  to  lines  should  be 
described  as  having  a   certain   direction.     Comparisons  with  the 


104  GRADED   ARITHMETIC.  [V.  61 

surface  of  still  water  and  with  a  plumb  line  will  lead  to  a  good 
description  of  horizontal  and  vertical  lines.  The  figures  given 
should  be  recognized  as  quadrilaterals,  and  the  reason  why  they 
are  quadrilaterals  should  be  given.  Parallel  lines  and  parallelo- 
grams, which  have  been  described  in  connection  with  exercises  in 
Book  IV.,  should  be  described  again. 

The  descriptions  called  for  in  4  may  be  given,  if  thought  advis- 
able, in  the  form  of  a  definition.  In  teaching  a  definition  the  object 
or  process  to  be  defined  should  be  brought  into  the  presence  of 
the  pupils,  and  by  questions  they  should  be  led  to  observe  those 
features  of  the  object  or  process  which  must  be  named  in  the 
definition.  For  example,  in  teaching  the  definition  of  a  square,  a 
square  surface,  preferably  of  a  cube,  is  presented.  The  pupils  are 
led  to  see  that  it  is  a  i^lane  figure  (that  having  been  previously 
taught),  that  it  is  hounded  hy  four  straight  lines,  that  the  sides  are 
equal,  and  that  it  has  four  right  angles.  From  these  facts  the 
following  statement  may  be  drawn  from  the  pupils  :  <'  A  square 
is  a  plane  figure  which  is  bounded  by  four  equal  straight  lines,  and 
which  has  four  right  angles."  If  a  rectangle  has  been  already 
defined,  the  pupils  might  say  that  the  square  is  a  rectangle  whose 
sides  are  equal.  Whenever  the  language  of  a  pupil's  definition  is 
faulty,  lead  him  to  see  the  faulty  or  bungling  construction  of  the 
definition  by  questions  or  by  comparison  with  a  correct  form. 

The  following  definitions  may  aid  teachers  in  drawing  from  the 
pupils  accurate  statements : 

A  rhombus  is  an  oblique-angled  parallelogram  whose  sides  are 
equal. 

A  rhomboid  is  an  oblique-angled  parallelogram  whose  opposite 
sides  only  are  equal. 

A  trapezoid  is  a  quadrilateral  which  has  only  two  sides  parallel. 

A  trapezium  is  a  quadrilateral  which  has  no  two  sides  parallel. 

61  Answers :  1  320  sq.  ft.  90  sq.  ft.  101^  sq.  ft.  3  yd.  4  in. 
42f  I  sq.  yd.  83^  sq.  ft.  123^  sq.  rd.  6  A.  40  sq.  rd.  2  A. 
137  sti.  rd.  189f  sq.  ft.     40  A.     2  A.  130  sq.  rd.     3^^  rd.     24^^  rd. 


V.  62]  teachers'  manual.  105 

19|  rd.       5542922.88  sq.  ft.       2  $25     |60     $156.25.        3  $378. 
4  $1158.75.       5  $7224.35.       6  $.005.       7  $1155.75  gain. 

5  may  be  performed  by  finding  first  the  square  chains,  and  then 
dividing  by  10,  the  niimber  of  square  chains  in  1  A.  Performed 
on  a  line,  the  solution  would  be  : 


$240  X  («M8^iil«) 


•    G2    A7isu'ers :   1  25  sq.  yd.     27  yd.       2  lOf  sq.  yd.     9^  sq.  yd. 

3  30  rd.  4  10  ft.  5  29^  yd.  6  7iJ  sq.  ft.  7  36f  sq.  ft. 
8  933^.  $11.66f.  9  30.  1451-.  10  $180  cheaper  by  sq.  ft. 
11  $5.31.  ^  12  3  10  6  20.  13  20  sq.  yd.  14  39.98+  A. 
15  $35.50. 

Nearly  all  of  these  problems  should  be  illustrated  by  plans.  In 
1,  assume  that  the  width  of  the  carpet  is  1  yd.  or  f  yd.  In  2, 
let  the  pupils  find  for  themselves  how  to  make  allowance  for 
corners.  A  properly-drawn  plan  will  give  all  needed  assistance. 
The  bricks  mentioned  in  12  are  of  common  size. 

63  Answers :    1    180  sq.  ft,  2    360  sq.  ft.         2640  sq.  ft. 

4  $2450.25.       5  $217.80.       8  2176  sq.  ft.       9  4416  sq.  ft. 

Finding  the  area  of  the  figure  in  1  is  a  review  of  what  has  been 
taught  previously,  but  the  hint  of  a  dotted  line  may  have  to  be 
given  before  the  pupils  can  find  the  correct  area.  Other  exercises 
similar  to  this  may  be  useful  for  practice.  The  pupils  will  dis- 
cover, when  they  compare  answers  to  4,  that  the  shape  of  the  lot, 
as  well  as  the  size,  determines  the  length  of  the  boi;ndary  line. 
To  solve  6,  the  given  polygon  must  be  divided  into  triangles  or 
rectangles,  and  the  needed  dimensions  be  determined  by  measure- 
ment. In  7,  let  the  pupils  estimate  the  dimensions  of  the  figures 
not  closer  than  the  sixteenth  of  an  inch. 

64  A  teaching  lesson  from  the  cube  should  precede  1.  The 
following  facts  should  be  observed  and  stated  by  the  pupils  :  "  The 
cube  is  a  solid.  (They  are  told  that  anything  which  has  length, 
breadth,  and  thickness  is  a  solid.)    It  is  bounded  by  six  faces.    The 


106  GRADED   ARITH]SIETIC.  [V.  65 

faces  are  equal  to  each  other.  Each  face  is  in  the  form  of  a  square." 
All  these  facts  may  be  gathered  into  one  statement  forming  a  defi- 
nition, thus  :  A  cube  is  a  solid  bounded  by  six  equal  square  faces. 

The  blocks  used  in  2  should  be  1-inch  cubes.  The  work  indicated 
in  3  should  be  performed  objectively  with  the  inch  cubes.  The 
rows,  layers,  and  number  of  layers  should  be  observed  by  the 
pupils,  and  the  number  of  cubic  inches  found  by  multiplying  the 
number  of  cubic  inches  in  a  row  by  the  number  of  rows,  and  this 
product  by  the  number  of  layers.  This  should  be  repeated  until 
it  is  well  understood. 

65  A7iswers :  1  60  cu.  in.  2  1728  cu.  in.  3  41472  cu.  in. 
4  20736  cu.  in.  5  4536  cu.  in.  2.6  cu.  ft.  6  4096  cu.  in. 
7  72  cu.  ft.  8  169  cu.  ft.  9  27  cu.  ft.  10  8  cu.  yd. 
20  cu.  yd.  3.51  cu.  yd.  11  14  cu.  yd.  22  cu.  ft.  23  cu.  yd. 
19  cu.  ft.  12  2  cu.  ft.  544  cu.  in.  6  cu.  ft.  1712  cu.  in. 
13  1152  cu.  in.  1296  cu.  in.  23328  cu.  in.  14  ^V  f- 
15  7  in.  by  5  in.  by  3  in.     105  cu.  in.     16  208  sq.  in.     142  sq.  in. 

There  are  two  ways  of  expressing  the  operation  of  changing  a 
number  from  one  denomination  to  another.  Thus,  in  10,  the  pupil 
may  be  led  to  say :  ''  There  is  |  as  many  cu.  yd.  as  there  are  cu. 
ft.";  or,  « There  are  as  many  cu.  yd.  in  216  cu.  ft.  as  there  are 
nines  in  216";  or,  "sls  many  cu.  yd.  as  9  cu.  ft.  is  contained 
times  in  216  cu.  ft."  But  in  all  such  explanations  care  should  be 
taken  that  the  pupil  sees  the  reason  for  the  step  taken.  Some- 
times it  is  well  in  such  exercises  to  ask  if  the  answer  is  to  be  less 
or  more  than  the  given  number,  and  then  to  ask  what  part  as  many 
or  how  many  times  as  many. 

66  A7iswers:  1  128  cu.  f t.  =  1  cd.  8  cd.  ft.  =  1  cd.  16  cu. 
ft.  =  1  cd.  ft.  3  64  cu.  ft.  48  cu.  ft.  96  cu.  ft.  4  80  cu.  fd. 
56  cu.  ft.  88  cu.  ft.  5  4  cd.  ft.  6i  cd.  ft.  24  cd.  ft.  6  i 
/^  f.  7  li  cd.  lt\  3f.  8  $3.12i.  9  396  cu.  ft.  24f 
cd.  ft.  3^\  cd.  10  $40|.  11  $136.64.  12  1248  packages. 
13  2027.5  papers, 


V.  67]  teachers'   JVIANtJAL.  107 

To  give  a  clearer  impression  of  a  cord  of  wood,  splints  4  in.  long 
might  be  piled  between  stakes.  The  pile  should  be  8  in.  long  and 
4  in.  high.  In  the  same  way  1  cord  foot  could  be  shown.  Piles 
of  other  dimensions  could  be  made  for  measurement.  The  pupils 
will  find  this  interesting  as  well  as  profitable  work. 

67  Ansivers:  1  227  p.  9051  p.  617  p.  2  30  sq.  rds.  3  6 
strips.  4  $76.56.  5  1023^  ft.  544^  ft.  6  150  ft.  22315-^ 
•sq.  rds.  7  $24.80.  8  60  posts  1530  pickets.  9  $13.22f 
529f  sq.ft.    $1.68.      10  198^bu.      11  1770^  sq.  ft.      12  $6.52^. 

In  the  first  part  of  1,  if  the  pickets  are  flush  with  the  ends  of 
the  fence,  one  space  will  be  only  -^  in.  wide.  Let  the  pupils  see 
this  by  dividing  the  space  in  inches  between  one  end  of  the  fence 
and  the  inside  edge  of  the  picket  on  the  other  end  by  3|-,  the 
number  of  inches  that  one  picket  and  one  space  occupy.  Make 
no  account  of  corners  in  the  last  part  of  1. 

68  Answers :  1  252^238^  cu.  ft.     2  |3f  a^  or  $3,878  +.     3  5§f  cd. 

4  30f §-!§§.  5  4^B/^;  16|ff  6  2H  cu.  ft.  7  32 J^  tons 
$176.34f. 

The  making  and  solving  of  original  problems  in  which  the  pupils' 
own  measurements  are  given  will  be  found  especially  valuable. 
Lead  the  pupils  in  such  work  to  apply  all  the  principles  that  they 
have  learned. 

69—70  For  information  as  to  U.S.  coins  see  Book  IV.,  page  89, 
and  note  thereon  in  the  Manual.  Most  of  the  problems  on  these 
pages  should  be  solved  orally. 

71    A7iswers:   1  $|.         2  $1.48+.         3  $117.         4  |78.62f 

5  $233.0625.  6  $123.12  gain.  7  9f  bu.  50  days.  8  $2.46. 
9  $51.84.  10  Brown,  $10.50  Eaton,  $8.79  Black,  $8,925 
Fuller,  $8.32  O'Connell,  $10.00  T.  Smith,  $10.74  A.  Smith, 
$11.42  Hall,  $11.70  Woods,  $11,375  Kelley,  $11.81  Reed, 
$8.49. 


108  GRADED   ARITH^SIETIC.  [V.  72 

Lead  the  pupils  to  employ  shorter  processes  -when  they  are 
easily  understood ;  thus,  in  1,  by  reducing  15  lb.  and  6f  lb.  to 
thirds  the  pupil  can  be  led  to  see  that  -y>-  will  cost  |  as  much  as 
4^5.  I  of  a  dollar  =  44f /.  The  pupils  should  be  told  that  gener- 
ally the  fraction  of  a  cent  is  dropped  if  it  is  less  than  half,  but 
that  in  this  and  all  similar  cases  the  extra  cent  would  probably 
be  charged.  Fractions  should  be  used  whenever  the  process  is 
shortened  thereby;  thus,  in  3  and  5:  6|-  M.  and  28;^  M.  should  be 
used  instead  of  6500  and  28250. 

72  A7iswers:  1  $8.50  $4.25  $1.70  $6.38  $15.86.  2  $62.50 
$625  $312.50  $156.25.  3  67^/.  4  48/  $38.40  $28.80. 
5  $1.50  $4.50  $5,625.  6  $175  $35  $140  $131.25.  7  $3.60 
$360  $180  $270.  8  $7.68  gain.  9  $4.20  gain.  10  $3f. 
11  $19^.  12  $20.34.  13  3i  lb.  274^  qt.  14  10.46  lb. 
15  Iff       16  223.25  dollars.       17  5  lb.  8|  oz. 

Most  of  these  exercises  can  be  performed  orally.  Lead  the 
pupils  to  work  through  100  or  10  when  more  convenient,  as  in 
Ito  7. 

73  Answers:  1  20.  2  1  lb.  42  oz.  3  144  2976  6096. 
4  31^  spoons.  5  57f  cu.  in.  6  11.63  qt.  7  2150.4  cu.  in.  537.6 
cu.  in.  14515.2  cu.  in.  6182.4  cu.  in.  8  17.14  qt.  9  348.348  bu. 
10  598.4  gal.  3111.9  gal.  72.3  gal.  11  18977.14  gal.  12  1386 
cu.  in.  larger.       13  37.23  qt.       14  1.6  qt. 

74  A?iswers:  5  72  sq.  rd.  12.8  sq.  rd.  5.6  sq.  rd.  261.33 
sq.  ft.  310  sq.  rd.  11  4  rd.  5  yd.  3  rd.  1  ft.  6  in.  17  yd. 
512^2  rd.  12  6  sq.  ft.  23  sq.  yd.  8  sq.  rd.  18  .25  rd.  .4  rd. 
19  .0625  mi.  .027  +  mi.  20  .1  A.  .0037  +  A.  29  .459  T. 
13600  lb. 

Have  as  many  of  these  performed  orally  as  possible.  A  few 
whose  answers  are  not  given  may  have  to  be  performed  by  the  aid 
of  figures.  For  the  sake  of  practice  in  Avritten  analysis  some  of 
the  easier  solutions  may  be  written  out  in  full,  as,  for  example : 


V.  75]  teachers'  manual.  109 


4 

.8  sq.  rd. 

30i 

2.42  sq.  yd. 

9 

9 

60)6200  see. 

3.78  s(i.  ft. 

60)103  min.  20  sec. 

144 

1  h.  43  min. 

1152 

Ans.:  1  h.  43  min.  20  sec 

1008 

112.32     sq.  in. 
Ans.:  2  sq.  yd.  3  sq.  ft.  112.32  sq.  in. 

75  Most  of  these  exercises  can  be  performed  orally,  advantage 
being  taken  of  convenient  fractions  whenever  possible. 

Frequent  practice  should  be  had  in  writing  receipts  and  bills. 

76  Answers:    2  Balance,  f 31.43.       3  Balance,  $89.25. 

Ask  the  pupils  to  explain  each  item  of  these  accounts,  and  lead 
them  to  distinguish  readily  which  items  belong  to  the  debit  side 
and  which  to  the  credit  side  of  the  account. 

Encourage  them  to  keep  a  cash  account  of  their  own  in  the  form 
given,  and  to  balance  the  account  at  the  end  of  every  week. 

77  A?mvers:  1  Balance,  $1063.45.  2  Balance,  $91.36. 
3  Balance,  $11.40. 

78  Point  out  the  difference  of  form  in  these  two  bills  of  sale, 
and  show  the  use  of  the  term  debtor  (Dr.)  in  the  second  bill.  Blank 
forms  are  easily  obtained,  and  yet  it  would  be  well  for  the  pupils 
to  be  able  at  any  time  to  rule  their  own  forms. 

79  ^wmws.-  1  Balance,  $166.07.  2  Balance,  $4.87^.  3  Bal- 
ance, $28,705.       4  Balance  due  June  1,  $23.24. 

Explain  fully  the  use  of  the  term  creditor  (Cr.).  Show  that  a 
man  is  debtor  for  all  that  he  receives,  and  that  he  is  creditor  for 
all  that  he  does  or  pays  out.  Give  several  exercises  illustrating 
this  point. 


110  GRADED    ARITHMETIC.  [V.  80 

80  Answers :  1  Balance  due,  $44.27.     2  Balance  due,  $137.35. 

3  S.  L.  Childs  owes  Asa  Howland,  $109.12,       4  $15.75,  one  man; 
$1890.       5  $7.90. 

In  the  last  answer  to  4  the  assumption  is  that  120  men  are 
employed  the  entire  time  given,  with  no  lost  time. 

81  A)isu-ers:    1    $15465.05.        2    $20208.14.        3    $14379.90. 

4  $12300.02.       5  11600     25200     696000     375000.       6  63000. 

5  to  10  should  be  performed  orally,  and  as  rapidly  as  possible. 
It  may  be  well  to  have  the  pupils  go  over  these  exercises  two  or 
three  times  for  the  sake  of  acquiring  facility  in  multiplying  or 
dividing  by  the  fractional  parts  of  10  and  100 

82  Ansivers:  20  $130.26.  21  $115.60.  22  $559.24. 
23  $297.15.     24  $165.84.     25  $249.84.     26  $9.55.     27  $108.54. 

The  iirst  19  exercises  are  for  quick  oral  practice.  In  multiplying 
by  a  number  two  or  three  units  less  than  100  or  1000,  lead  the  puj^ils 
to  make  two  multiplications  and  subtract ;  thus,  in  4  :  93  X  100  = 
9300 ;  93  X  2  =  186  ;  9300  —  186  =  9114.  After  a  little  practice 
such  sokitions  may  be  made  silently,  the  answer  only  being  given. 
In  9,  first  multiply  by  10,  30,  and  60,  and  add  to  the  products  the 
multiplicands.  Let  the  pupils  perform  orally  as  many  of  the 
exercises  from  20  to  27  as  they  can. 

83  A?iswers:   1  $1.92.     2  $21,675.     3  $118,625.      4  $42.98. 

5  $81.15.  6  $6.75.  7  $4.59.  8  $11.66f  Total,  $289.35||f 
9  9.5.  10  1.853i|a.  17  $90666.66|.  18  $10125,  val.  of 
estate     $5625,  wife     $1125,  son. 

84  Answers:  3  31  A.  139i|i  sq.  rd.  4  Gj\  lb.  5  2^ 
sq.  rd.    1  A.  70  sq.  rd.     6  $31.68.     8  207J.     9  287' f     10  $6.53^. 

12  55/.     $11.00.       $54 

85  Answers:  1  26  poles.  2  70  posts.  3  99  posts.  4  1782 
pickets.  5  2970  stones.  6  396^9^  sheets.  7  $6.00.  8  $7.55+. 
9  145  trees.        10  3-^j  s(i.  rd.        11  146fgor;   sq.  rd.        12  8|f. 

13  31-^7.  14  69^}.  15  6|§.  16  8^/.  17  800  doz.  18  $25.00. 
19  131.84  pts. 


V.  86]  teachers'  manual.  Ill 

The  pupils  may  be  able  to  perform  some  of  these  problems- 
orally.  Let  plans  be  drawn  for  all  problems  in  mensuration. 
When  the  steps  of  a  process  are  not  clearly  seen,  make  the  reason- 
ing clear  by  questioning,  and  use  small  numbers. 

86  Answers:  1  24.99  sq.  rd.  2  .025  A.  3  $16.02.  4  28001b. 
$13104.  5  $240.07i.  16  844if  planks.  17  2150.4  cu.  in. 
18278.4  cu.  in. 

87  Answers:  1  4.821  bu.  2  336.294  bu.  20177.64  1b.  3^ 
400  ft.  37i  ft.  10  $9,074-  $16,335  $12.10.  11  46|  cu.  yd. 
12    463  lb.  5i  oz.     10502f  lb.       13  f . 

88  Aiisivers:  6^  U  1300  1b.  1700  1b.  $10.  7  $857} 
$1714f  $3428i.  8  5184  lb.  9  128  flagstones.  10  1536 
bottles.         11  29.403  lots.         12  166§  yd.         13  45000  oranges. 

14  $62487889.60. 

89  A?iswers:  1  1  mi.  109  rd.  6^  ft.  2  li||  sec.  3  231  rd. 
3i  ft.  4  59  lb.  6  oz.  5  750  lb.  6  18600  ft.  3||  mi. 
7  3  mi.  220  rd.  5/^  ft.  8  1200  times.  9  ij  week.  8^\^  hr. 
10  Mf-        11  '^2.50     3tV  yd.        12  llllfl  cu.  ft.        14  $9.53. 

15  $1.50     $3.75     $2     $2.30. 


SECTION   VIII. 

NOTES   FOR   BOOK   NUMBER    SIX. 

Before  taking  up  the  exercises  included  in  this  book,  the  pupils 
are  supposed  to  have  a  tliorough  knowledge  of  common  and  deci- 
mal fractions  and  of  some  of  the  simpler  and  more  important 
processes  in  mensuration  and  denominate  numbers.  If  these  sub- 
jects are  not  well  understood,  it  is  advised  that  a  more  extended 
review  of  Book  V.  be  given  than  is  found  in  Section  I. 

It  is  not  unlikely  that  some  parts  of  mensuration  may  be  thought 
too  difficult ;  but  if  the  previous  work  has  been  well  done  and  each 
new  principle  is  taught  objectively,  there  will  be  no  difficulty  which 
it  will  not  be  well  for  the  pupils  to  overcome.     In  the  applications 


112  GRADED   ARITHMETIC.  [VI.  1 

of  percentage,  also,  which  include  all  kinds  of  problems  that  are 
likely  to  occur  in  the  every-day  life  of  a  mechanic  or  farmer,  there 
will  be  some  difficulties  both  in  the  understanding  of  principles 
and  in  the  processes  involved  unless  the  foundation  is  carefully 
laid.  The  development  exercises  preceding  the  practical  problems 
should  be  carefully  and  systematically  followed  if  good  results  are 
to  be  expected. 

As  the  work  progresses  less  attention  need  be  given  to  the 
mechanical  operations  with  numbers  if  the  previous  work  has  been 
thoroughly  done,  while  a  constantly  increasing  emphasis  should  be 
put  upon  the  solution  of  problems  which  require  close  and  careful 
thinking.  Increased  attention  also  should  be  given  to  the  explana- 
tion of  processes  and  the  formulation  of  rules  and  definitions. 
With  this  changed  work  of  the  pupils  comes  a  corresponding 
change  in  the  character  of  the  teaching  and  drilling.  While  the 
use  of  objects  and  drawings  in  making  clear  a  new  process  or  prin- 
ciple is  always  desirable,  there  is  not  so  complete  dependence  upon 
them  as  the  pupils  come  to  have  a  fuller  power  of  generalization 
and  reasoning. 

Other  hints  of  a  general  nature  will  be  found  in  the  Note  to 
Teachers,  which  precedes  the  first  section  of  the  book. 

1  Answers:  1  1490.  2  1392.  3  1528.  4  1402.  5  1588. 
6  1448.  7  1534.  8  1371.  9  1537.  10  1362.  11  633. 
12  769.  13  607.  14  664.  15  624.  16  612.  17  492. 
18  599.  19  474.  20  644.  21  608.  22  714.  23  532. 
24  611.  25  535.  26  471.  27  591.  28  672.  29  624. 
30  673.       31  700.       32  471.       33  757.       34  548. 

Encourage  the  pupils  to  add  by  pairs,  thus  (1) :  14,  26,  47,  etc. 
As  a  convenience  it  may  be  well  to  have  the  sum  of  each       ^.^ 
column  placed  below  the  line.     For  example,  the  partial  and     -i  ok 
total  sums  in  1  would  appear  as  follows  :  1490 

2  A7iswers:  1  $49.97.  2  $50.56.  3  $61.93.  4  $55.44. 
5  $55.87.  6  $56.48.  7  $54.69.  8  $46.20.  9  $51.34. 
10  $49.50.      11  $46.79.      12  $35.01.      13  $41.15.     14  $47.63. 


VI.  3]  teachers'    IVIANUAL.  113 

15  $34.44.  16  32.98.  17  $384.94.  18  $385.04. 

19  $2506.82.  20  $2188.85.  21  $2352.38.  22  $1710.47. 
23  $1905.60.  24  $1216.46.  25  $2395.32.  26  $258.41. 
27  $2101.83.  28  $1177.55.  29  $1376.14.  30  $2138.41. 
31  $10864.12.  32  $10664.12.  33  $848.32  $340.39  $788.07 
$389.81  $508.91.  34  $62.84  $355.95  $888.56  $381.19 
$574.05.  35  $315.83  $611.81  $0.44  $676.29  $339.05. 
36  $323.71  $600.23  $318.33  $49.61  $269.06.  37  $427.83 
$664.71  $331.89  $525.47  $36.59.  38  $84.73  $665.29 
$650.43  $30.34  $722.74.  39  $1642.54  $2100.91  $2191.90 
$1691.54       $1830.77.  40    $2490.86       $1760.52       $1403.83 

$1291.73       $1321.86.  41    $2428.02       $2116.47       $2292.39 

$1672.92       $1895.91.  42    $2112.19       $1504.66       $2291.95 

$1096.63       $1656.86.  43    $2435.90       $2104.89       $1973.62 

$1146.24       $1825.92.  44    $2008.07       $1440.18       $2305.51 

$1671.71  $1862.51.  45  $776.34  $412.37  $6.42  $289.56 
$13.04  $249.92  $500.10.  46  $72.54  $520.22  $12.39 
$623.76  $294.80  $701.80  $613.92.  47  $131.55  $529.81 
$22.44  $553.41  $185.47  $8.11  $688.88.  48  $45.90  $165 
$27.86  $265.10  $484.55  $4.33  $757.41.  49  $703.80 
$932.59       $18.81        $334.20        $307.84        $451.88        $113.82. 

50  $59.01     $9.59     $34.83     $70.35     $480.27     $709.91     $73.96. 

51  $85.65     $364.81    $50.30    $288.31     $299.08     $3.78     $68.ry:^. 

52  $835.35  $402.78  $41.25  $219.21  $493.31  $459.99 
$575.06.  53  $352.18  $2379.36  $179.61  $1707.20 
$1106.63  $877.39  $1554.93.  54  $1128.52  $1966.99 
$186.03  $1417.64  $1093.59  $627.47  $2055.03. 
55  $1055.98  $1446.77  $198.42  $2041.40  $798.79 
$1329.27  $1441.11.  56  $1187.53  $1976.58  $220.86 
$1487.99       $613.32       $1337.38       $2129.99. 

To  add  in  lines  as  well  as  in  columns  is  good  practice,  since 
additions  have  to  be  so  made  sometimes  in  business. 

3    Ansu-ers:    1   $295946306.08.         2    $20,456+.         9   715920 
1909120  14318400  9545600  71592000  214776000. 


114 


GRADED  ARITHMETIC. 


[VI.  4 


10  1232240 

10628070. 

477146360. 

878640320. 

38944325. 

714319200. 

4697111. 

1578fi. 

14611-ff 

4970  4  9  ■'> 

'±Vt       V^Q  Jg^. 

200261t\V 


1509494    24644800  26185100 

11  1622538   50659242  58651744 

12  37552900  72472220  59401860 

13  2335200   38190250  42082250 


26369936 
516808400 
690583200 

34487985 
5882220190 


9698|§ 
31782^1 


6070751811 


14  569754600     539211785     496692785 
15  1127461     72157|i     9019/^0     474143 

16  3000281     197386§|     62505^-     7653' 

17  125000      80000      158730i§      22727  iSj-      42016//^ 
18  572572f     1336001     83500^     318095f     79682^^% 

19    1894779tV      947389i§      761818|f      535481/^ 
151117t*vV-  20     103621421^1  8232936f§ 

47323171 1}       13976S3|§       146729^^^. 

Problems  similar  to  the  following  might  be  given  from  the 
table  :  What  was  the  difference  in  amount  of  imports  in  1891  and 
1892  from  England  ?  from  France  ?  from  Germany  ?  from  Brazil  ? 
from  Mexico  ?  from  Cuba  ?  What  was  the  difference  in  exports 
in  1891  and  1892  to  England  ?  to  France  ?  etc.  By  how  much 
did  the  exports  to  England  exceed  the  imports  from  that  country 
in  1891  ?  in  1892  ?  What  can  you  say  of  the  difference  between 
the  exports  to  France  and  the  imports  from  that  country  in  1891  ? 
in  1892?  etc.  Find  the  total  imports  in  1891  from  England, 
France,  Germany,  Brazil,  Mexico,  and  Cuba ;  etc. 


4    A7isivers:    1  25ff. 


195|f 


13- 


1143331^2^4 


Let  the  pupils  practice  upon  exercises  in  cancellation  until  they 
can  recognize  readily  tlie  common  factors  of  dividend  and  divisor. 
Answers  to  5  and  6  should  be  given  as  rapidly  as  possible  without 


the  aid  of  figures. 

5    Ans^vers:   52  603^^,.        53  681*3.        59  317. 

60  49tV 

61  543  1.     62  107§§.     63  106i|.      64  m.     65  46||. 

66  693 

67  60^3_.       68  29i. 

In  1  to  20  lead  the  pupils  first  to  find  l)y  inspection  the  least 
common  denominator,  and  then  to  add  as  the  reduction  is  made. 


6   Ansivers : 

■   1  lll^VV         2  13 

03- 

3U      4      39f. 

12  800      9      7 

1( 

2i     li.       16 

G4     6}^     4^     46f. 

23  27.0763. 

24  1827.8249. 

Vi.  6]  teachers'  manual.  115 

Thus,  in  10,  the  pupils  would  say:  "The  least  common  denomi- 
nator is  12  ;    ^9^  +  /j=14;    11  +  ^2^  =  ^1;    H  +  T\  =  tl;    tl  +  A 

=  fl  =  2i" 

Possibly  21  to  31  may  have  to  be  performed  by  the  aid  of 
figures,  but  the  pupils  should  be  given  an  opportunity  to  try  their 
solution  orally. 

3  3p.         11  &%      21 

3  5.    13  ii,v  iM  n 

21  13.9352.      22  627.3294. 

7  For  the  sake  of  practice  in  pointing  off,  the  solution  of  1  to  7 
may  be  written  out.  All  the  remaining  exercises  on  this  page 
should  be  performed  orally. 

8  All  these  problems,  with  the  possible  exception  of  1,  2,  11, 
and  13,  should  be  performed  orally.  Let  careful  attention  be 
given  to  the  pupils'  explanations. 

9  Nearly  all  of  these  review  problems  should  be  performed 
orally. 

10  Ansivers:  1  100  rd.  34f  rd.  3  fgoo.  4  5  rd.  3  ft. 
10.56  in.  5  $13.93+.  16  32670  sq.  ft.  17  5080|  sq.  ft. 
18  6076  sq.  in.  19  1^  A.  6^  A.  ^  A.  20  fW  A.  ^%%%  A, 
21  If  f  f  f       22  $298.07.       23  6352^  sq.  ft.     $1030.75. 

11  Ansivers:   1  $4420.80     318  yd.      4  $3840     $2400     $512. 

In  places  where  this  method  of  measurement  is  common,  many 
exercises  similar  to  3  and  4  should  be  given.  Descriptions  of  lots 
of  land  found  in  deeds,  advertisements,  etc.,  should  be  brought  into 
the  class  and  fully  exj^lained. 

13    Ansivers:   5  32  yd.  breadthwise.        6  29  yd.  6  in.      27  yd. 

8  in.         7  f  ft.      i  ft.      1|-  in.         8  60^^^  sq.  ft.       IG^f  courses. 

9  10  sq.  ft.     20  sq.  ft.       10  32  sq.  ft. 

If  necessary  use  strips  of  paper  to  illustrate  5  and  6.  Several 
illustrations  similar  to  that  given  in  12  will  assist  the  pupils  to 
discover  how  the  average  width  of  a  board  may  be  found,  and  in 
general  how  the  area  of  any  trapezoid  may  be  found. 


116  GEADED    ARITHMETIC.  [VI.  13 

13  Answers:  7  26}|  sq.  ft.  8  23yVg\  sq.  rd.  9  160  sq.  ft. 
10  11^  sq.  yd.  11  142f  sq.  yd.  12  384  sq.  ft.  13  576  sq.  ft. 
113^  sq.  yd. 

The  description  of  plane  figures  here  called  for  should  be  more 
carefully  made  than  that  made  in  answer  to  the  same  question 
in  Book  V.  (page  57).  Tlie  description  should  be  as  nearly  as 
possible  in  the  form  of  an  accurate  definition.  The  pupils  should 
be  led  first  to  see  that  a  plane  surface  is  a  surface  in  which  a  line 
connecting  any  two  points  will  lie  wholly  in  the  surface,  and  that 
a  plane  figure  is  a  portion  of  a  plane  surface  bounded  by  one 
or  more  lines.  These  lines  may  be  straight  or  curved.  Figures 
bounded  by  straight  lines  are  rectilinear  figures,  and  figures  bounded 
by  curved  lines  are  curvilinear  figures.  In  teaching  the  definitions 
of  the  various  figures,  first  present  the  figure  and  call  attention  by 
questions  to  its  essential  characteristics.  For  example,  after  pre- 
senting a  triangle,  the  teacher  says:  "What  kind  of  a  figure  is 
this  ?  By  how  many  sides  is  it  bounded  ?  Define  triangle."  The 
pupil  is  led  to  say:  "A  triangle  is  a  plane  rectilinear  figure  bounded 
by  three  sides."  If  his  wording  of  the  definition  is  not  quite  as 
good  as  it  should  be,  lead  him  by  questioning  to  make  the  desired 
correction.  The  following  definitions  may  be  developed  in  the 
same  way: 

A  polygon  is  a  plane  rectilinear  figure  having  three  or  more 
sides. 

A  quadrilateral  is  a  polygon  of  four  sides. 

A  pentagon  is  a  polygon  of  five  sides.     Etc. 

A  square  is  a  right-angled  parallelogram^  having  its  sides 
equal. 

A  rectangle  is  a  right-angled  parallelogram  having  only  its  op- 
posite sides  equal. 

A  rhombus  is  an  oblique-angled  parallelogram  having  its  sides 
equal. 

1  This  term  is  supposed  to  have  been  taught  previously.     See  Manual,  p.  76. 


VI.  14]  teachers'  manual.  117 

A  rhomboid  is  an  oblique-angled  parallelogram  having  only  its 
opposite  sides  equal. 

The  walk  mentioned  in  13  is  supposed  to  be  included  in  the 
garden.  If  tlie  blackboards  in  the  room  be  of  the  same  width, 
lead  the  pupils  to  find  first  the  entire  length  of  all  the  boards, 
and  then  to  multiply  by  the  width. 

14  Answers:    2  $10.40.       3  $60     $44.       4  43^  sq.  ft. 

After  the  pupils  have  drawn  plans  illustrating  the  solution  of 
3,  lead  them  to  show  how  they  can  always  find  by  a  short  process 
the  length  of  a  walk  or  trench  if  made  on  the  inside  border  of 
a  given  rectangular  lot,  and  also  its  length  if  made  on  the  outside 
of  the  lot.  To  find  the  length  of  a  walk  if  made  on  the  inside 
border  of  a  rectangular  lot,  subtract  four  times  the  width  of  tlie 
walk  from  the  perimeter  of  the  lot ;  and  to  find  the  length  of  a 
walk  if  made  on  the  outside  of  such  a  lot,  add  four  times  the  width 
of  the  walk  to  the  perimeter  of  the  lot. 

In  the  solution  of  4,  lead  the  pupils  to  connect  the  corners  by 
dotted  lines,  and  to  find  by  measurement  the  altitude.  That  the 
altitude  may  be  as  exact  as  possible,  see  that  the  line  measured  is 
exactly  perpendicular  to  the  base.  The  square  edge  of  a  card  will 
be  a  good  means  of  measurement.  Let  the  figures  made  for  5  be 
exchanged  among  members  of  the  class  and  answers  be  compared. 
Let  the  radius,  angles,  etc.,  be  designated  by  letters  ;  thus,  ao,  aob, 
etc. 

15  Slow  and  careful  measurements  are  needed  here.  Do  not 
take  up  a  new  exercise  until  the  pupils  thoroughly  understand  the 
jDrevious  one.  The  points  to  be  developed  in  each  of  the  exercises 
are  as  follows  :  1.  The  angles  at  the  centre  of  a  circle  are  measured 
by  the  arcs  which  subtend  them.  2.  A  simple  way  of  finding  angles 
of  various  magnitudes.  3.  To  make  a  protractor  for  measiiring 
angles.  Let  this  be  most  carefully  made.  4.  Practice  in  esti- 
mating magnitude  of  angles.  5.  The  sum  of  all  the  angles  made 
about  a  given  point  is  equal  to  360°.  6.  The  opposite  angles  made 
\)j  two  lines  crossing  each  other  are  equal.     7.  The  sum  of  the 


118  GRADED    AEITHJMETIC.  [VI.  16 

three  angles  of  a  triangle  is  equal  to  two  right  angles.  The  sum 
of  the  angles  of  a  polygon  is  equal  to  two  right  angles  taken  as 
many  times  as  the  figure  has  sides  less  two. 

It  is  not  desirable  at  this  time  for  the  pupils  to  spend  much 
time  in  formulating  the  facts.  The  important  point  is  for  them 
to  discover  the  facts  by  observation  and  construction. 

16  Answers:  6  258  sq.  ft.  7  143.86+  sq.  rd.  8  46.7694 
sq.  rd.       9  Exact  area  =  2781.1584  sq.  ft. 

Proceed  very  slowly  with  the  first  four  exercises,  and  review 
frequently  so  that  the  meaning  of  the  terms  may  be  fixed  in  the 
mind. 

The  polygon  abode  is  a  regular  polygon,  because  it  has  equal 
sides  and  equal  angles.  The  j^upils  are  supposed  to  have  com- 
passes in  constructing  the  figures  mentioned  in  3.  These  two 
ways  of  constructing  regular  polygons  should  be  repeated  until 
they  are  well  understood. 

From  the  questions  and  illustrations  of  4  the  pupils  can  easily 
discover  the  process  of  finding  the  area  of  any  regular  polygon. 
The  pupils  should  be  told  that  the  line  fo  is  the  apothem  or  less 
radius.  The  rule  for  finding  the  area  of  a  regular  polygon  should 
be  made  by  the  pupils,  and  may  be  as  follows  :  Multiply  the  pe- 
rimeter by  half  the  apothem,  or  multiply  half  the  perimeter  by  the 
apothem.  The  exact  length  of  the  apothem  of  the  regular  pen- 
tagon mentioned  in  5  is  8.2584  ft.  (.6882  X  12).  The  measured 
distance  will  be,  of  course,  only  approximately  correct.  The  ratio 
of  the  apothem  to  the  side  of  a  regular  polygon  is  as  follows  : 
triangle,  .2887;  pentagon,  .6882;  hexagon,  .866;  octagon,  1,2071; 
decagon,  1.5388  ;  dodecagon,  1.866. 

17  Answers:  1  $5628.  2  Neither  71^  yd.  3  12f  yd. 
49.6  yd.  28|  yd.  4  53^  sq.  yd.  37f  sq.  yd.  21^  sq.  yd.  21^ 
sq.  yd.       5  611  sq.  ft.     25  rolls. 

Let  the  pupils  carefully  measure  the  lines  whose  distances  are 
not  indicated.  According  to  the  measurement  given  of  the  hall, 
one  breadth  of  the  carpet  will  just  fill  the  narrow  hall-way.     Four 


VI.  18]  teachers'  manual.  119 

strips,  therefore,  will  be  needed,  one  strip  20  ft.  long  and  3  strips 
each  6  ft.  long. 

In  estimating  the  amount  of  paper  that  ■will  be  needed  for  the 
walls,  only  an  approximate  estimate  can  be  made,  as  more  or  less 
may  be  needed  for  matching,  etc.  A  good  method  is,  first  to  find 
the  number  of  strips  required,  then  to  find  how  many  strips  can  be 
cut  from  a  roll.  The  whole  number  of  strips  required  divided  by 
the  number  that  can  be  cut  from  a  roll  will  give  the  number  of 
rolls  required.  Let  the  pupils  perform  6  by  this  method.  The 
pupils'  answers  to  6  can  be  but  ajiproximately  near  the  correct 
answers  without  definite  instructions  as  to  how  the  wainscot  will 
be  placed  with  reference  to  chimney,  and  whether  the  window  in 
the  bay  extends  below  the  wainscot.  The  widest  part  of  the 
recess  of  the  bay  window  is  supposed  to  be  10  ft.  wide,  and  the 
narrowest  part  4  ft.  wide.     Depth  of  bay,  4  ft. 

The  given  floor  plan  may  be  xised  for  other  exercises,  such  as 
finding  the  number  of  sq.  ft.  of  flooring  in  each  room,  the  cost  of 
painting  a  border  for  the  floor  of  each  room,  etc. 

18  ^nsM-ers;  3  314  sq.ft.  4  40.13  sq.  rd.  .2508+  A.  5  1925 
sq.  ft.  8  314.16  sq.  ft.  1256.64  sq.  ft.  9  3.183  ft.  15.9154  ft. 
2.758  yd.  10  1.76+  sq.  ft.  9.621+  sq.  ft.  11  420.168+ 
times. 

Let  the  pupils  try  both  ways  of  finding  the  area  of  a  circle. 
The  rule  for  finding  the  area  which  the  pupils  should  be  led  to 
give,  from  the  questions  in  1,  is  :  "  Multiply  the  circumference  by 
half  the  radius."  And  it  is  by  this  rule  that  the  pupils  should  at 
first  be  required  to  find  the  area  of  a  circle.  3.1416  may  be  used 
as  the  ratio  of  the  circumference  to  the  diameter. 

19  Answers:  2  384  sq.  in.  3  7^^|  sq.  ft.  4  17568  sq.  in. 
5  $180     12926?f  gals. 

The  cube  and  other  right  prisms  should  be  presented  to  the 
class  in  teaching  these  exercises.  Right  prisms  only  should  be 
dealt  with  at  first,  and  when  prism  is  referred  to  in  this  book,  a 
right  prism  is  meant.     The  points  of  resemblance  in  all  prisms 


120  GRADED   ARITHMETIC.  [VI.  20 

are  :  (1)  The  two  bases  are  parallel.  (2)  The  lateral  sides  are 
parallelograms.  The  definition,  therefore,  of  a  prism  which  may 
be  made  by  the  pupils  is  :  "A  solid  whose  bases  are  parallel  and 
whose  lateral  sides  are  parallelograms." 

Only  the  approximate  contents  can  be  given  in  answer  to  the 
questions  asked  in  8  and  9,  owing  to  the  fact  that  the  exact  dimen- 
sions of  the  prisms  cannot  be  found  from  the  cuts.  In  explaining 
the  process  of  finding  the  contents  of  prisms,  lead  the  pupils  at 
first  to  use  the  method  given  on  page  106  of  the  Manual. 

20  Answers :  1  192  cu.  in.  364  cu.  in.  2  24|  cu.  ft.  3  75 
sq.  ft.  27.06  cu.  ft.  4  24  sq.  ft.  330  cu.  ft.  5  311f  sq.  ft. 
3888  gals.  8  3  sq.  ft.  25.78+  sq.  in.  618.72+  cu.  in.  9  .319+ 
cu.  in.     5.1051  sq.  in. 

The  work  called  for  in  1  should  be  done  very  carefully.  If  the 
models  are  made  of  cardboard,  let  the  pupils  first  cut  the  outline 
as  given,  and  then  cut  half  through  the  cardboard  where  the  edges 
are  to  be.  The  figures  when  folded  can  be  held  in  place  by  means 
of  mucilage  or  paste.  Modeling  may  be  continued  with  profit  to 
the  pupils,  besides  furnishing  models  for  future  measurements. 

The  following  facts  concerning  the  cylinder  should  be  taught 
objectively  and  definitions  developed  : 

A  cylinder  (right  cylinder)  is  a  solid  bounded  by  a  curved  sur- 
face and  by  two  equal  parallel  circles.  The  curved  surface  is  called 
the  convex  surface,  and  the  other  two  sides  are  called  the  bases. 

To  find  the  convex  surface  of  a  cylinder,  multiply  the  circum- 
ference of  the  base  by  the  height. 

To  find  the  volume  of  a  cylinder,  multiply  the  surface  of  one 
of  the  bases  by  the  height. 

31  Answers:  1  $40.72+.  2  $113.10-.  3  706.86  cu.  in. 
4  1800  cu.  in.  5  in.  5  8  in.  6  10  ft.  7  2  ft.  8  U  ft. 
9  ir,  ft.  8  in.     10  3911.38+  in.  11  8f  sq.  yd.      12  $17.79-. 

13  ^  cu.  ft.  14  3630  cu.  ft.  15  640  sq.  ft.  $9.60  $5.12. 
16  171^  yd. 


VI.  22]  teachers'  manual.  121 

After  solving  several  simple  problems  similar  to  5,  let  the  rule 
be  given  for  finding  one  dimension  of  a  solid  when  the  cubic  con- 
tents and  two  dimensions  are  given.  Whenever  any  point  is  not 
clear,  use  models  and  small  numbers.  Let  the  pupils  illustrate 
all  processes  by  drawings  when  it  can  be  done. 

33  Answers:  1  56  sq.  yd.  62*  sq.  yd.  24  yd.  2  40  ft. 
320  sq.  ft.  3  486  sq.  ft.  54  sq.  yd.  4  35^  sq.  yd.  5  8  rolls. 
8  104  sq.  ft.  12f  sq.  yds.  9  272  sq.  ft.  $13.60.  10  18  yd. 
11  26f  yd.     widtlnvise  of  the  room. 

For  method  of  finding  the  approximate  number  of  rolls  of  paper 
required  to  paper  the  walls  of  a  room,  see  Manual,  page  119.  Let 
5  be  performed  by  that  method.  Great  care  should  be  taken  in 
the  solution  of  all  these  problems  to  desig- 

18 

nate  what  each  concrete  number  stands  for. 
Another  good  way  is  to  indicate  the  proc- 
ess in  one  place  and  perform  the  indicated 
operations  in  another  place.  For  example, 
in  8. 

[(16  X  2)  +  (10  X  2)]  X  2  =  sq.  ft.  in  border. 


q' 


(18  X  12)  -  104 

•^^ ~ =  sq.  yd.  m  fioor  unpainted. 


32 
20 


52 
lOl  sq.  ft.  ^^^  ''I-  y^- 


216 
104 
9)112 


or, 


(18  X  12)  —  [(18  -  4)  X  (12  -  4)]  =  sq.  ft.  in  border. 

14 

14  X  8  ,   .    ^  .  ,  ^  216  8 

sq.  yd.  m  floor  unpainted 


9  ^  '    '  ^  112  9)112 

104  sq.  ft.  12f  sq.  yd. 

Either  of  these  solutions  or  any  other  correct  solution  should 
be  accepted,  and  may  be  followed  by  an  oral  explanation. 


122  GRADED    ARITHMETIC.  [VI.  28 

If  the  pupils  are  not  considered  ready  for  such  work  as  that 
noted  above,  they  should  be  led  to  write  out  the  solution,  so  as 
to  indicate  that  they  understand  every  step.  The  following  is  a 
sample  of  what  should  be  expected: 

18  18  ft.  —  4  ft.  =  14  ft.,  length  of  unpainted  part. 

12  12  ft.  —  4  ft.  =    8  ft.,  width   "         "  " 

216  sq.  ft.  in  entire  floor.  14 

112  ''     "    "  unpainted  part.         __8 

104  "     "    "  border.  112  sq.  ft.  in  unpainted  part. 

9  sq.  ft.)112     sq.  ft. 

12|  =  no.  sq.  yd.  in  unpainted  part. 

23  Ans7rers :  1  20^  bd.  ft.  2  19  iV  bd.  ft.  3  96  ft.  4  2652  ft. 
$42.43.  5  1421^  bd.  ft.  $21.32.  6  12/^  loads.  7  1564^ 
loads.  8  $376.32.  9  2.9+  pailfuls.  10  172j  173.57+  bu. 
144  bu.  5  T.  8  cwt.  11  4700.16  gal.  12  34.27+  rd.  93.46+ 
sq.  rd. 

34  Answers :  1  23^  in.  2  5  ft.  6.84  in.  3  36  sq.  ft.  4  70^ 
sq.  ft.       5  $56.10     $3234.33. 

Dictation  exercises  similar  to  6  will  be  found  interesting  and 
profitable  to  the  pupils.  Original  problems  of  the  same  kind 
should  be  made  and  brought  into  the  class. 

35-37  If  the  measures  have  been  properly  used  in  previous 
grades,  they  need  not  be  used  in  the  solution  of  these  problems, 
most  of  which  should  be  performed  orally.  Let  the  pupils  perform 
the  problems  in  the  shortest  and  most  direct  way.  Correct  and 
lead  by  questioning  rather  than  by  giving  outright  the  shorter 
method.  For  example,  if  the  pupils  attempt  to  perform  24, 
page  26,  in  a  long  or  roundabout  way,  ask  such  questions  as  : 
"What  short  way  of  getting  the  cost  of  25  bu.  at  75/  a  bushel? 
(Probably  two  ways  will  be  given  :  one,  \  of  100  times  75/,  and 
the  other,  25  times  f  of  a  dollar.)  How  much  is  it  ?  3  pk.  6  qt. 
is  how  many  quarts  less  than  a  bushel  ?  What  part  of  a  bushel 
is  it  ?  3  pk.  6  qt.,  then,  will  cost  how  much  less  than  75/?" 
After  answering  these  questions  the  pupils  can  easily  perform  the 


VI.  28]  teachers'  manual.  123 

problem,  and  will  say:  "1  bu.  costs  f  of  a  dollar  ;  25  bu.  will  cost 
25  times  f  of  a  dollar  or  ^^,  equal  to  $18.75.  3  pk.  6  qt,  is  2  qt. 
or  ^^  of  a  bushel  less  than  a  bushel,  and  will  therefore  cost  ^^  of 
75/  less  than  75/.  ^^  of  75/  is  4  cents  and  a  fraction,  subtracted 
from  75/  equals  71/.  This  sum  added  to  $18.75  is  $19.46,  the 
cost  of  25  bu.  3  pk.  6  qt.  of  wheat  at  75/  a  bushel." 

28  —  30  If  possible,  bring  to  the  class  the  various  coins  referred 
to  in  these  exercises,  and  talk  about  their  absolute  and  relative 
values.  Some  of  the  solutions  may  need  the  aid  of  figures,  but 
most  of  them  can  be  performed  orally.  Information  in  regard  to 
coins  not  contained  here  will  be  found  in  the  Appendix  of  Book 
VIII. ;  also  on  page  89,  Book  IV.,  and  page  69,  Book  V.,  and 
notes  thereon  in  the  Manual. 

31    Answers:    10  12  1b.  4  oz.         11  12  cwt.  20  1b.        12  6  T. 

10  cwt.  30  lb.  13  16  T.  5  cwt.  50  lb.  14  1  lb.  7  pwt.  15  4  lb. 
2  oz.  2  pwt.  16  2i.  17  9 /§  4/3  lOm.  18  8  cwt.  91  lb. 
13  oz.  9  T.  18  cwt.  8  lb.  12  oz.  19  101  2  3  29  6/§  16iTl. 
20  4  T.  18  cwt.  21  3  T.  11  cwt.  82  lbs.  22  9  oz.  4  pwt. 
23  2S  13  23  17  gr.  24  43  29  5  gr.  7  lb.  7  oz.  3  dr.  1  sc. 
25  1  T.  1  cwt.  60  lb.     3  T.  12  cwt.  20  lb. 

33  Answers:  1  4  lb.  51.  2  9/§  4/3.  3  27  T.  3  cwt. 
60  lb.       4  8  lb.  1  oz.  15  pwt.       5  9  T.  17  cwt.  2  lb.  8  oz.     112  T. 

12  cwt.  80  lb.  6  38  lb.  19  10  gr.  148  cong.  4  0.  7  19  cwt. 
551b.  8|  oz.     8  3  oz.  10  pwt.  3^gr.     9  3  3  29  5  gr.     10  2  3  16ni. 

11  9  cwt.  89  lb.  5i  oz.     5  oz.  2  pwt.  18  gr.      12  537.5  lbs.     $8.06. 

13  3  oz.  15  pwt.  12  gr.  14  l\  T.  15  |gj  lb.  16  $240. 
17  $2.05.  18  1/5  2/3.  19  7  lbs.  1  oz.  16  pwt.  16  gr. 
71  lb.  6  oz.  6  pwt.  16  gr.  20  $89.08.  21  200  pills.  360  pills. 
22  28,  and  37  pwt.  left. 

33  Answers  :  2  J  640  rd.  K  1920  rd.  L  960  rd.  M  1280  rd. 
N  320  rd.  0  2880  rd.  P  2560  rd.  Q  2240  rd.  3  J  87120 
sq.  ft.  K  261360  sq.  ft.  L  130680  sq.  ft.  M  174240  sq.  ft. 
N  43560  sq.  ft.  0  392040  sq.  ft.  P  348480  sq.  ft.  Q  304920 
sq.  ft.        4  </  48  qt.     K  60  qt.     L  40  qt.     M  52  qt.     N  24  qt. 


124  GRADED   ABITHMETIC.  [VI.  33 

0  32  qt.  P  44  qt.  Q  50,  qt.  S  J  U  qt.  K  192  qt.  L  96  qt. 
M  128  qt.  N  32  qt.  ,  0  288  qt.  P  256  qt.  Q  224  qt.  6  J^  16s. 
K  12s.  L  5s.  M  15s.  N  Is.  0  2.5s.  P  7.5s.  Q  12.16s. 
7  J  1320  gv.  K  840  gr.  L  1860  gr.  M  3264  gr.  i\r  2700  gr. 
0  2320  gr.  P  3640  gr.  Q  1696  gr.  8  J  j%\  A.  /f  ^3^9/^  A. 
L  ^\S  A.  M  ^^^  A.  AT  ^V'.i^  A.  0  if^S%  A.  P  ^WsV  A. 
^  t¥A  a.  Q  J  ^%%  T.  a:  IV^x  T.  L  2i^  T.  Jf  2j-«_6_  t. 
AT  2/A«^  T.  0  3^ ef  0  T.  P  1  J'^s.T^^  T.  Q  3^%  t.  10  J  880  yd. 
JT  1320  yd.  L  1100  yd.  TIf  733^  yd.  N  10261  yd.  0  938|  yd. 
P  1496  yd.  Q  1114f  yd.  11  J  232^  cd.  A^  4961  cd.  L  5010  cd. 
Jf  672  cd.  A^  563f  cd.  0  151\  cd.  P  496f  cd.  Q  876  cd. 
12  J  960  gi.  a:  1920  gi.  A  2880  gi.  Ji"  1168  gi.  A^  3344  gi. 

0  5792  gi.  P  5088  gi.  Q  4320  gi.  13  J  14520  ft.  AT  9240  ft. 
L  20460  ft.  Jf  35904  ft.  A^  29700  ft.  0  25520  ft.  P  40040  ft 
Q  18656  ft.  14  ./  25.6  qt.  K  19.2  qt.  L  8  qt.  J/  24  qt. 
X  1.60  qt.  0  4  qt.  P  12  qt.  Q  19.456  qt.  15  J  320  A. 
A  480  A.  L  400  A.  J/  266  A.  106  sq.  rd.  1811  sq.  ft.  A^  373  A. 
53  sq.  rd.  2721  sq.  ft.  0  341  A.  63  sq.  rd.  272^  sq.  ft.  P  544  A. 
Q  405  A.  53  sq.  rd.  272^  sq.  ft."  16  J  1000  lb.  K  1500  lb. 
L  1250  lb.  3f  833  lb.  5^  oz.  X  1166  lb.  lOf  oz.  0  1066  lb. 
10|  oz.  P  1700  lb.  Q  1266  lb.  lOf  oz.  17  J^  80  lb.  X  60  lb. 
i  25  lb.  M  75  lb.  A^  5  lb.  0  12.5  lb.  P  37.5  lb.  Q  60.8  lb. 
18  J  256  rd.  a:  192  rd.  L  80  rd.  Jf  240  rd.  A^  16  rd.  0  40  rd. 
P  120  rd.  Q   194  rd.  9  ft.  2.88  in.  19  ^  11  qt.  K  7  qt.  L   15  qt. 

1  pt.  M  27  qt.  If  gi.  A^  22  qt.  1  pt.  0  19  qt.  2f  gills.  P  30  qt. 
2|  gi.  ()  14  qt.  1  tV  gi-  20  ^  54  oz.  K  75  oz.  A  52  oz.  16  pwt. 
3f  73  oz.  10  pwt.  A"  101  oz.  2  pwt.  17.28  gr.  0  76  oz.  12  pwt. 
15.36  gr.  P  121  oz.  1  pwt.  14.4  gr.  Q  216  oz.  1  pwt.  4.8  gr. 
21  J  13  cu.  ft.  864  cu.  in.  K  20  cu.  ft.  432  cu.  in.  A  16  cu.  ft. 
1512  cu.  in.  31  11  cu.  ft.  432  cu.  in.  A"^  15  cu.  ft.  1296  cu.  in. 
0  14  cu.  ft.  6911  cu.  in.  P  22  cu.  ft.  1641|  cu.  in.  Q  17  cu.  ft. 
172|cu.  in.  22  c/52rd.  A' 75  rd.  A  42  rd.  31 91  vd.  A^52rd. 
0  60  rd.  P  84  rd.  Q  82  rd.  23  J  133  T.  A'  258  T.  10  cwt. 
L  236  T.  31  346  T.  16  cwt.  A^  271  T.  9  cwt.  0  354  T.  1800  cwt. 
P  271  T.  1400  cwt.  Q   418  T.  800  cwt.   24  J  162  rd.  4  yd.  4.^  in. 


VI.  34]  '  teachers'  manual.  125 

K  241  rd.  4  yd.  4^  in.     L  203  rd.  4  yd.  2  ft.  5^  in.     M  140  rd. 

2  ft.  2f  in.  N  192  rd.  1  yd.  1  ft.  9f  in.  0  175  rd.  2  yd.  2  ft.  3  in. 
P  279  rd.  3  yd.  7|  in.  ()  206  rd.  2  yd.  2^  in.  25  .7"  13  A.  100 
sq.  rd.  if  30  A.  130  sq.  rd.  L  25  A.  80  sq.  rd.  M  37  A.  96  sq.  rd. 
N  29  A.  29  sq.  rd.  C)  46  A.  138  sq.  rd.  P  32  A.  134  sq.  rd.  Q  50  A. 
124  sq.  rd.  26  J  1  bn.  3  pk.  5  qt.  1.2  pt.  K  2  bn.  5  qt.  0.4  pt. 
L  1  bu.  1  pk.  ;;  (^t.  0.4  pt.  M  2  bu.  1  pk.  1  qt.  N  2  bn.  5  qt. 
0.048  pt.  O  1  bu.  2  pk.  7  qt.  0.176  pt.  P  2  bu.  3  pk.  4  qt.  1.44  pt. 
Q  5  bu.  ;5  (jt.  0.992  pt.  27  ./  16  en.  ft.  432  cu.  in.  K  22  cu.  ft. 
L  20  cu.  ft.  1296  cu.  in.  M  18  cu.  ft.  86.4  cu.  in.  N  21  cu.  ft. 
648  cu.  in.  0  19  cu.  ft.  403.2  cu.  in.  P  1  cu.  yd.  3  cu.  ft.  3240 
cu.  in.  Q  20cu.  ft.  1094.4  cu.  in.  28  A  4  rd.  B  3rd.  C  20  rd. 
D  120  rd.  E  2110  rd.  F  \  rd.,  or  4  ft.  U  in.  ^  0.2  rd.,  or  3  ft. 
3.6  in.  H  1  rd.  /  1.75  rd.,  or  1  rd.  4  yd.  4^  in.  29  A  5  gal.  1  qt. 
B  12  gal.  2  qt.        C  52  gal.       i)  150  gal.      E  2950  gal.      P  2  qt. 

3  gi.  6^  2  qt.  1.2  gi.  ZT  3  qt.  1  gi.  7  5  gals.  1  pt.  0.8  gi. 
30  J  40  S(i.  rd.  a:  60  sq.  rd.  L  68  sq.  rd.  M  11^  sq.  rd. 
N  47ird.  0  33^  sq.  rd.  P  63  sq.  rd.  Q  33^  S(i.  rd.  31  ./  2  bu. 
a:  0  pk.  L  2  bu.  M  6  bu.  2  pk.  N  5  bu.  2  pk.  0  5  bu.  P  7  bu. 
1  pk.  Q  3  bu.  32  ./  123.5  sq.  rd.  K  89.75  sq.  rd.  L  35.6 
sq.  rd.  M  113.875  sq.  rd.  N  428  sq.  rd.  0  13.614  sq.  rd.  P  49.91 
sq.  rd.  Q  79.275  sq.  rd.  33  ./  7^^.  K  IS^s.  L  ^s.  M  l^%s. 
N  Q^^^s.  0  5,f  ,s-.  P  9tV.  Q  9^2_5.  34  j  l^?^  qt.  K  11 -^\  qt. 
L  11/t-  qt.  J/  15/^  qt.  N  6^4  qt.  0  9iJ  qt.  P  12f  qt. 
Q  16if  qt.  35  ./  f  180.  K  $375.  Z  $140.80.  M  $477.75. 
N  $387.69-.  0  $332.07+.  P  $736.57.  Q  $1224.34. 
36  J  $17.88-.  A'  $11.38-.  L  $25.19-.  M  $44.20. 
AT $36.56+.  0  $31.42-.  P  $49.29+.  <)  $22.97-.  37-7$30.61+. 
JT  $45.92-.  X  $38.27-.  Jf  $25.51-.  A^  $35.72-.  0  $32.65+. 
P  $52.04+.     Q  $38.78-. 

34  Answers:  1  Apr.,  June,  Sept.,  Nov. ;  Jan.,  Mar.,  May,  July, 
Aug.,  Oct.,  Dec;  Feb.  2  365  52  wk.  1  da.  366  da.  4  -$546 
or  $547.75.  5  4  mo.  7  da.  6  mo.  10  da.  2  mo.  6  da.  6  T  yr. 
1  mo.  29  da.     22  yr.  2  mo.  29  da.        7  400  da.     473  da.     298?  ia. 


126  GRADED   ARITHMETIC.  [VI.  35 

8  $342.08.  10  67  yr.  9  mo.  22  da.  11  April  15,  1865. 
12  Goethe  :  82  yr.  6  mo,  23  da.  Longfellow,  75  yr.  0  mo.  25  da. 
Luther,  62  yr.  3  mo.  8  da.  Franklin,  84  yr.  3  mo.  0  da.  Shake- 
speare, 52  yr.  Napoleon  I.,  51  yr.  8  mo.  20  da.  Burns,  37  yr. 
5  mo.  26  da.     Milton,  65  yr.  10  mo.  30  da. 

The  number  of  days  in  each  month  of  the  year  should  be  re- 
peated until  it  can  be  told  as  soon  as  the  name  of  the  month  is 
given. 

A  good  method  of  finding  the  months  and  days  from  one  date 
to  another  is  illustrated  by  the  following  solution  of  5  :  From 
Aug.  13  to  Dec.  13  is  4  mo.;  from  Dec.  13  to  Dec.  20  is  7  da. 
Answer:  4  mo.  7  da.  From  June  30  to  Aug.  30  is  2  mo.;  from 
Aug.  30  to  Sept.  5  =  1  day  in  August  +  5  days  in  Sept.  =  6  da. 
Ansiver:  2  mo.  6  da.  And  in  12  :  From  1749  to  1831  is  82  yr.; 
from  Aug.  28  to  Feb.  28  is  6  mo.;  from  Feb.  28  to  Mar.  22  is 
22  da.     Ansu-er :  82  yr.  6  mo.  22  da. 

A  common  custom  in  banks  is  to  regard  30  days  for  every  month 
in  estimating  time.  For  example,  in  finding  the  time  between 
June  30  and  Sept.  5  the  process  would  be  :  From  June  30  to 
Sept.  30  =  3  mo.  —  25  da.  (the  time  from  Sept.  5  to  Sept.  30)  = 
2  mo.  5  da. 

In  performing  the  latter  part  of  4,  let  the  pupils  find  the  amount 
saved  if  the  year  begins  on  a  week-day,  and,  again,  if  the  year 
begins  on  Sunday;  also  if  the  year  is  a  leap  year. 

To  find  the  exact  number  of  days  from  one  date  to  another  (7) : 
From  Sept.  16,  1891,  to  Sept.  16,  1892,  is  366  da.;  14  more  days 
in  Sept.  +  20  in  Oct.  =  34  da.,  which,  added  to  366,  equals 
400   da. 

35  Answers:  4  378'  754'.  5  1820"  2558".  6  10848" 
14424".  7  5°  17°  7^°.  8  .3  of  a  degree  .6  of  a  degree 
.275  of  a  degree        §|°.  9  22°  12'.  10  New  York,  74°  W. 

Baltimore,   76°  35'  W.  Chicago,   87°  35'  W.  San    Francisco, 

122°  25'  W.  11  16°  33'  30".  86I3IJ5+  "^i-  13  21600  geog.  mi. 
24840  statute  mi. 


VI.  36]  teachers'  manual.  127 

Lead  the  pupils  to  use  the  globe  and  outline  map  in  estimating 
distances.  Show  by  means  of  the  globe  the  reason  for  the  differ- 
ence in  length  of  the  degree  of  longitude  in  various  latitudes. 
Some  of  the  distances  are  given  in  the  Appendix  of  Book  VIII. 

36  Answers:  1  192  mi.  12.5  hr.  5  da.  5  hr.  2  21  da.  If  hr. 
3  312  lb.  8  oz.  4  3  T.  750  lb.  5  537.605  cu.  in.  G7.201  cu.  in. 
6  19.286  bu.  7  247.76  bu.  8  198.2  bu.  9  97.71+  bu. 
10  64.28+  qt.  11  (a)  1.75  M.  (b)  3.50  M.  (c)  1.29/  (d)  1.29/ 
(e)  25.8/     (/)  14.902. 

For  all  practical  purposes,  in  estimating  the  capacity  of  bins, 
the  bushel,  stricken  measure,  may  be  regarded  as  containing  1:|- 
cu.  ft.  The  answers  above  given  are  found  by  exact  measure. 
Show  that  a  bin  or  box  will  hold  f  as  many  bushels  by  heaped 
measure  as  it  will  by  stricken  measure,  on  the  supposition  that 
the  heaped  measure  will  hold  ^  more  than  the  stricken  measure. 

37  Answers:  13^^/.  2  $1104.  8  $65.75  gain.  9  $12. 
10  $120.  11  7560  bricks  648  bricks  less.  12  158  rd.  6.6  ft. 
13  7H  loads.       14  733^  loads. 

The  solution  of  problems  on  a  line  sometimes  shortens  the  pro- 
cess, as  shown  in  the  following  solution  of  1 : 

.60 
$.00 

$6.00  ,,  ,,     „,,,       $0.00X^X3_^^,^ 

49 

The  tons  mentioned  in  8  are  short  tons.  Performed  on  a  line, 
the  solution  is  :  -j^qq 

$5.25  X^g^- ($5.20X128.3). 

112 
Bricks  vary  greatly  in  size,  the  dimensions   here  used  for  11 
being  8^"  X  4"  X  2^".     Let  the  pupils   see   how  the   bricks   must 
be  placed  in  both  instances  to  make  a  pile  of  the  given  dimensions. 


128  GRADED   ARITHJVrETIC.  [VI.  38 

Tlie  common  bricks  are  placed  on  edge,  and  the  Milwaukee  bricks 
are  laid  flat. 

38    A7isicers :   1  4  T.  1800  lb.  2    70|§  bbl.         183f  |  bbl. 

89if  bbl.  3  17.5  T.  4  25  T.  1440  lb.  5  11600  lb.  59^%  bbl. 
6  $36.80.  7  $2.73^5.  8  3824i  mi.  9  9  A.  87  sq.  rd.  24 
sq.  yd.  6i  sq.  ft.  10  160  rd.  6  mi.  230  rd.  11  36  mi.  226f  mi. 
141  mi.  12  95f/.  13  556  bu.  $444.80  $300.24.  14  114f  lb. 
2  bu.  1J|  qt.       15  34  cu.  ft.  1483xV  cu.  in. 

It  is  assumed  in  7  that  the  flour  bought  is  whole  Avheat  flour. 

31)  In  addition  to  the  helps  here  given,  all  the  common  meas- 
ures of  the  metric  system  should  be  used  for  the  exercises  of  this 
section.  Much  time  should  be  given  to  actual  measurements  before 
the  exercises  are  taken. 

If  a  meter-stick  is  not  provided,  each  pupil  can  easily  make  one 
by  cutting  a  stick  exactly  ten  times  as  long  as  the  decimeter  shown 
in  the  cut.  The  stick  can  be  divided  by  cross-lines  into  decimeters 
and  centimeters. 

40  Answers:  7  8.64^™  86.4 1^™  864^^"'.  9  8570.704°^. 
10  9030.043™.  11  312.5  times.  12  2280'".  13  $5.74. 
14  70601  poles. 

Similar  exercises  may  be  given  for  class  drill  if  needed.  Show 
two  ways  of  reading  metric  numbers  ;  thus,  69.83™  may  be  read 
sixty-nine  and  eighty-three  hundredths  meters,  or,  sixty-nine  meters 
and  eighty-three  centimeters.  Let  the  pupils  practice  in  reading 
numbers  in  both  ways. 

41  Answers:  2  10000 ^«™.  3  1000000 1™.  5  .0001 1 ^''" 
.000001  qJ^'"  .001  "Km_  e  10001™  lOO^™  lOO^™.  7  9.603<)76i'^"\ 
8  8.697544  <i™.  11  6849602.1 1™.  12  5005000.8 1™. 
13  251.125 <i™.  14  2000001™.  15  12500  bricks.  16  5* 
.5^^     500^     .05  ^       17  200*     4=^     .08 '^     6000  ^ 

The  sign  for  square  is  sometimes  written  sq.  instead  of  q.  Call 
attention  to  the  fact  that,  as  100  units  of  one  denomination  make 
a  unit  of  a  denomination  next  larger,  two  places  of  figures  are 
allowed  for  each  denomination. 


VI.  42]  teachers'  manual.  129 

43  Answers:  1  $407.50.  2  $500.  3  $320  gain.  4  20001™ 
20 ^  6  0.001  <■»'«       0.000001*^"'"       0.000000001'""'.         7  Oue 

thousand   times    as    large.  8     One    million    times    as    large. 

9  8048000™'™.  100.0080'""'.  11  357*="">        142.8  loads. 

12  1.4476  +  ''""\       13  120 «'.       14  58.5 «'.       15  $294.84. 

43  Aiiswers:  3  800 1  GO^  0.8 ^  4^  900 1.  4  S'^''  0.78 '^'^ 
4'^''.       5  450 '»!     0.8 '^1     50-11     400OO'".      6  0.02 1.      7  0000 ^     GO"'. 

The  liter  measure  is  one  decimeter  long,  wide,  and  deep.  If  the 
school  is  not  supplied  with  a  liter,  one  can  be  made  of  wood  or 
tin.  This  measure,  filled  with  water  at  a  temperature  a  little 
above  freezing,  weighs  a  thousand  grams,  or  one  kilogram.  These 
facts  should  be  given  to  show  the  relation  that  the  measures  of 
volume,  capacity,  and  weight  have  with  one  another,  all  depending 
upon  the  meter,  which  is  supposed  to  be  the  ten-millionth  part  of 
the  distance  on  a  meridian  from  the  equator  to  the  pole. 

44  Avswers:   1  400^'     8000  s     9000 «    80 «    G()s.      2  834.971 «. 

3  G09.80lg.         5   8G.97S       84  s.         6   80^.         1000^.         8   l^^". 
9  12000  Ks.       10  24000  i^X 

The  last  four  exercises  on  the  page  are  important  only  as  they 
serve  to  give  tlie  pupils  an  idea  of  the  value  of  the  common  metric 
measures  in  measures  familiar  to  them.  It  may  not  l)e  necessary 
for  all  these  exercises  to  be  given  to  accomplish  the  purpose. 

45  Anstvers:  2  2|  hr.  3  $200  $21.50.  4  1341f"'. 
5  13.88'^'"'.      7  36753G0K".      g  2323.2^1.      9  720001.      ^.0  25.8  «^ 

11  5"'.  12  .04502411^'"       $36019.20.  13  71GGf'""       $48. 
14  399700'!'".       15  goO"'. 

46  Aiisivers:    1    $20.34.  2  270001.         3  S.G'^'*™        3G00i. 

4  51.072'^"'"     31.92  loads.        5  10  hr.        6  l^^X        7  4297.674 'i"! 
42.97674 ^  8  9^*.         9  4.8"\         10  3 1' 447 1^»^.         11  12.5i^X 

12  7112+  half-dollars.       13  45208.8 s.       14  $7188.      15  30.35 ^ 
One  important  end  should  be  gained  in  work  upon  the  metric 

system,  and  that  is,  to  give  pupils  a  clear  idea  of  the  simplicity 
of  the  system  in  comparison  with  our  system  in  common  use,  and 


130  GRADED   ARITHMETIC.  [VI.  47 

the  great  saving  of  time  which  its  adoption  would  occasion.  If  it 
is  desired  that  pupils  shall  be  acquainted  with  the  denominations 
of  the  system  Avell  enough  to  perform  problems  at  any  time,  fre- 
quent reviews  will  be  necessary. 

47—49  Follow  carefully  the  order  of  exercises  here  given,  and 
let  the  pupils  practice  upon  them  until  they  can  perform  them 
readily.  Their  understanding  of  subsequent  work  Avill  depend 
largely  upon  the  thoroughness  with  which  they  take  this  prelim- 
inary work. 

50  Answers:  1  0.696  2.7130f  75.1152.  2  296.1  447.43^ 
0.033026.  3  923.68  45.08  44.54^.  4  $35.8234  17.4147^ 
2550.62^.  5  $44.889i  1134.3392  886.033  mi.  6  2000.882 
$154.26^        142.47^.  7  0.0039501       0.21375       47.66625  A. 

8  9.945  0.5899  29.41.  9  0.01^^  ^.437  0.684988.  10  2.80^ 
0.0027.  0.0001369|.  19  1.852  0.03538  0.002f.  20  $64.8284 
1.3008i  0.07^f  21  23.465  0.3962425  0.06/^.  22  0.063615 
0.002^^  35.6325.  23  22633.65  1980.68  55.8723f. 
24  $0.17226  $16.25525  0.004f.  25  0.003^  217.875 
0.0033432^.  26  2102.118  hr.  0.012^6^  15.772328.  27  43.6720f 
40559.44f.     0.247f       28  0.05^     0.000032     19630. 

51  Nearly  all  of  these  problems  should  be  performed  orally, 
but  for  the  sake  of  learning  a  good  form  of  written  analysis,  the 
pupils  might  write  out  the  solution  of  some  of  them.  The  following 
form  of  solution  is  suggested  : 

$3550.00     cost  of  farm. 

.334- 
14  Given  :  Cost  of  farm.  $1183  33t^ 

Required:  The  selling  price.  'irrr\ac\ 

$4733.33i  selling  price. 

53  Ansivers:  1  $6375.  2  $.0807.  $.1521.  3  $1456. 
4  $5098.     5  3835201b.    940  bu.     6  97f  bu.     7  $630.     8  $84.81. 

9  3031.2  bbl.  10  38250.  11  $21^  loss.  12  $359.13. 
13  $49.93. 


Yl.  63]  tejlchers'  manual.  131 

Lead  the  pupils  in  such  problems  as  13  to  shorten  the  process 
as  much  as  possible.  In  this  problem  they  Avould  say:  "There  are 
332  gallons  in  all.  If  8*}^  of  the  molasses  was  lost  and  45"^  of  it 
was  sold,  there  would  be  left  47%  of  it.  47%  of  332  is  156.04 
gallons.  32  cents,  the  price  of  a  gallon,  multiplied  by  the  number 
of  gallons  is  $49.93." 

53  Answers:  1  7  cwt.  9.5  lb.  2  )$2523.17.  3  1.64|  yd. 
4  $112.50.  5  $1827.50.  6  $117.  7  $2100000.  8  787828.29f 
tons.  9  640485.449  tons.  10  1099962.  11  1515111.54. 
12  $1450.80.       13  6  lb.       14  $43.80. 

When  these  problems  have  been  solved  and  solutions  in  good 
form  written  out,  the  pupils  might  practice  in  writing  the  solution 
on  a  line,  using  hundredths  for  per  cent,  as  in  12,  as  follows  : 

93  2 

$3  X  XH(t>  X  M    ,  $4  X  I860  X  60       «.-,,. ^^^ 

li — ^< — m + ~5 — X — m  ^  ^^^^^•^^• 

54—55  A  new  principle  is  involved  in  these  exercises.  In 
previous  exercises  the  base  and  rate  were  given  to  find  the  per- 
centage. In  these  which  follow,  the  base  and  percentage  are  given 
to  find  the  rate.  These  terms  need  not  be  given  to  the  pupils  at 
present.  They  should  be  led  to  recognize  the  conditions  after 
sufficient  practice.  Whenever  the  rate  per  cent  that  one  number 
is  of  another  is  called  for,  let  the  pupils  first  get  the  common  frac- 
tional part  and  then  reduce  the  fraction  to  liundredths,  or  per 
cent.  Great  care  should  be  taken  to  lead  the  pupils  to  recognize 
the  base,  or  the  number  of  which  another  number  is  a  part.  The 
form  of  question  should  be  variously  given,  so  that  the  pupils  will 
not  work  mechanically.  Sometimes  the  base  is  the  first  number 
given  in  the  question,  as,  What  part  of  8  is  4  ?  and  sometimes 
the  percentage  is  the  first  number  given,  as,  4  is  what  part  of  8  ? 
Sufficient  practice  in  both  forms  should  be  given,  so  as  to  enable 
the  pupils  to  recognize  the  base  ifhererer  it  is  placed  in  the 
question. 


132  GRADED   ARITHMETIC.  [\1.  56 

• 

56  Answers:   1  12.5%    S.GO^i-r^    197i|§%    16&Ui%    25/^%. 

2  67.5%  |f%  46t\%  11%  34713%.  3  lJ^J^j%  f|% 
14^%  Ml%  114111%.  4  40/^%  2^y,%  266^1^%  29|gf% 
4§ix%.  5  24ff%  131%  18i%  12^,%  39-}!.  6  l§if% 
90x|«%  30if%  5^U%  33^h-  7  30|e%  389^i,%  74if^,% 
554i|M%  562ifff%.  8  «  12^%;  ^-111%;  c  33%;  .7  12|%; 
e7i%;/88%;  yl2i%;  A  20%;  i  i%;  J  U^.  9  a  6%;  h  30%; 
c7i%;  fZ5%;  e50%;/16%;  ^74%;  A  15//g  % ;  fl5%;il4f%. 
10  6|%  26|%  200%  800%  13^%  15i;f%  7-}3.%  n^cf^ 
15t\V^%  Hm%-  11  50%  35%  28%  80411%  93^% 
100%  38|%  10i|%  5%  71%.  12  36i%.  13  Canada, 
5MMU%;  U.S.,  69H^§H%     14|%. 

57  Ansivers:  1  Mass.,  .22+%  KY.,  1.36%  Cal.,  4.45+% 
Texas,  7.49+%.        2  Moh.,  5.^,+  %     Brah.,  20-if%     Bud.,  40%. 

3  650%.  4  Col.,  13+%  Ind.,f+%  Whites,  86.1+%.  5  87.3+%. 
6  1.4+%.  7  5%.  Q^^%S%.  9  14.2%.  10  15%.  1119.2%. 
12  li%. 

Notice  the  wording  of  3.  Ask  the  pupils  to  give  examples  of 
both  kinds  of  exercises. 

The  term  base  might  be  introduced  here  as  being  the  amount  on 
■which  the  rate  is  estimated.  Some  introduction  will  have  to  be 
given  to  show  what  the  base  is  in  such  exercises  as  7,  8,  9,  and  11. 

5S  On  this  page  and  the  two  following  pages  are  simple  appli- 
cations of  the  two  principles  of  percentage  already  taught.  The 
pupils  should  perform  the  problems  by  analysis,  recognizing  in  the 
varying  conditions  what  principle  is  to  be  applied.  Of  the  two 
methods  of  finding  the  interest  of  a  given  sum  for  a  period  greater 
or  less  than  a  year,  it  is  better  on  some  accounts  to  find  the  interest 
first  for  one  year,  and  then  for  the  given  time,  as  indicated  in  8. 
Let  the  analysis  be  simple  and  direct,  no  set  form  being  required. 

50  By  ''rate  of  interest,"  referred  to  in  1,  the  pupils  should 
know  that  the  rate  per  year  is  meant.  The  pupils  should  be  told 
that  tlie  per  cent  of  gain  or  loss  in  business  transactions  is  always 
estimated  on  the  cost  unless  otherwise  specified.     Some  talk  about 


A"I.  60]  teachers'  manual.  133 

insurance,  commission,  savings  banks,  etc.,  will  be  helpfiil  in  con- 
nection with,  problems  bringing  in  those  terms.  For  facts  concern- 
ing these  subjects  see  Book  VII.,  and  accompanying  notes  in  the 
Manual. 

60  Some  of  these  problems  will  need  to  be  solved  by  the  aid 
of  figures.  The  analysis  of  the  most  difficult  ones  may  be  written 
out  if  thought  best. 

61  In  these  exercises  the  base  is  asked  for.  Analysis  in  the 
use  of  common  fractions  should  first  be  made,  such  as  :  ''  Four  is 
one  half  of  some  number  ;  two  halves  of  the  number  is  two  times 
four,  or  eight";  and  again:  "Twenty-four  is  two  thirds  of  some 
number  ;  one  third  of  the  number  is  one  half  of  tAventy-four,  or 
twelve.  If  one  third  of  the  number  is  twelve,  three  thirds  of  the 
number  is  three  times  twelve,  or  thirty-six." 

From  such  reasoning  the  step  to  similar  work  in  percentage  will 
be  easy.  The  pupils  will  readily  see  that  ^^J'^,  or  50%,  may  be 
treated  exactly  as  ^  is  treated,  and  that  12  is  50"^  of  2  times  12. 
When  the  required  number  cannot  be  obtained  directly  by  multi- 
plying or  by  using  an  equivalent  common  fraction  for  the'  given 
per  cent,  1%  of  the  nunVber  is  first  found,  and  then  100*^;  thus, 
in  10,  the  pupil  will  say:  "Six  dollars  is  six  per  cent  of  some 
number  ;  one  per  cent  is  one  sixth  of  six  dollars,  or  one  dollar. 
If  one  per  cent  of  the  number  is  one  dollar,  one  hundred  per  cent 
of  the  number  is  one  hundred  times  one  dollar,  or  one  hundred 
dollars." 

A  similar  analysis  should  be  made  in  solving  all  the  problems 
on  this  page.  To  assist  in  making  the  solution  clear,  the  analysis 
may  be  written  out ;  thus,  in  13  : 

j\<\y  of  the  number  =  32 

100  4  0    Ui  o^ 

|o  0    a      u  u  =  j_o_o   of  32  =  80 

32  is  4:0%  of  80. 

62  Use  the  common  fractional  form  in  the  solution  of  1;  and 
generally,  when  the  pupils  find  difficulty  in  solving  a  problem  in 


134  GRADED    ARITHMETIC.  [VI.  63 

percentage,  let  them  substitute  small  numbers  in  place  of  large 
ones,  and  common  fractions  in  place  of  percentages.  The  substi- 
tution of  aliquot  parts  should  be  made  for  percentages  whenever 
simplicity  is  gained,  but  for  useful  practice  some  of  the  percentages 
on  this  page  should  be  used  ;  thus,  in  4,  the  analysis  may  be  : 
"l§o=the  required  number;  AS  =  ^%%  ;  t^tt  =  ^n  of  ^5,  or  ^; 
igo  =  100  times  -J,  or  50."  And  in  8  :  "  \^^  —  the  required  num- 
ber. $40  =  i§5;  ^^^  =  ^^^  of  $40,  or  32/;  |n  =  100  times  32/, 
or  $32." 

63  A7isivers:  1  308^  2155|  $6493  $380800  $12006.25. 
2  £42500  1739^^^  tons  2409||tons  365  da.  19405|.  3  $11552.94 
$8183^  $14028.55+  $8728.88+.  4  88|.  5  214|.  6  420.  . 
7  822|.  8  405f  9  345.  10  105.  11  10800.  12  120. 
13  111^.  14  2741^  15  28000.  16  252.  17  H  18  100. 
19  93.98+.  20  $2517.482+.  21  $2500.  22  1225  bii. 
$2858.33^.  23  $113.35.  24  $265.07.  25  $155928.88  +  . 
26  531i.       27  $4.04,  cost  per  bbl. 

The  following  form  of  written  analysis  for  these  exercises  is 
suggested  : 

21         10  0  =  cost  of  house. 

1^6  :=:  selling  price  of  house  =  $400. 

,       .       ,      $400 
T^o  of  cost  =  ^^. 

,nn      f  ^         $400    X   100         ^...Q„ 

|o  0  of  cost  =  — j-^ =  $o44.83. 

64  The  miscellaneous  exercises  on  this  and  the  following  pages 
of  the  section  involve  all  the  principles  of  percentage  which  have 
been  taught.  Do  not  give  the  pupils  any  direct  assistance  in  their 
solution ;  nor  should  any  assistance  be  given  until  the  pupils  have 
made  a  strong  effort  to  do  the  work  by  themselves.  If  the  condi- 
tions of  a  problem  are  not  understood,  make  them  clear  by  judicious 
questioning. 


VI.  65]  teachers'  manual.  135 

65  Ajiswers:  1  352.83^.  2  39 /j.  3  306.  4  1.41. 
5  42  lb.  2f  oz.          6  1%.          7  0.525.         8  100%.         9  180%. 

10  0.01G65.  11  4900.  12  25.  13  |2.247.  14  4if%. 
15  TSlTj^  men.  16  3.G9G.  17  774|f.  18  55^^\%. 

19  87^%.  20  3140f.  21  5375.  22  14904.  23  1100. 
24  83(;Ubii.  25  tIst-  26  74/3^^%.  31  ^236^  $1111^. 
32  $8250         $18250.  33  1^/  per  lb.         $22.50   entire  gain 

20  lb.  for  $1. 

66  Ansirers:  1  2634.016.  2  $5.40.  7  $3  40%.  8  15% 
gain     15%  loss.       9  6.62+%.       13  65ia-f/.       15  $300. 

67  Ansicet-s:    1  $8888|     $13333^.      2  361-6/.      3  5023.14|f. 

4  Lose  $50.       5  $653.33^.       6  $129.16f. 

68  Ansivers:   1  30%.         2  $4.         3  $13403/^.        4  $6.11^. 

5  $970.31.  6  $160.  7  $10.  8  2^%.  9  $1400  $7000 
$3500  $14000  $9333^  10  $118.81^.  11  $1237.50. 
12  $6000.  13  $2115  pre.  $2500,  Lon.  $3333^,  Hon. 
$2000,  Boston     $2166|,  KY. 

69  A?isiaers:  1  Snm  paid,  $7500  greater.  2  $12828.58. 
3  $115,  pre.  4  $356.25,  cost  of  insurance.  5  $8.10,  com. 

6  $37,  com.  $1443,  returned.  7  $40  collected  25%. 
8  $400,  cost.  9  $20,  com.  10  $1969.80,  cost.  11  $3900, 
cost.     $29^,  com.       12  9.36%. 

70  Ansivers:  1  60%.  2  108  lb.  64±  lb.  3  $5.80  on 
$1000  $.0058  on  $1.  4  $240.  5  4000  1b.  7  82f%. 
8  2000  lb.  wood      37|  lb.  ashes.         9  4400.97+  lb.         10  83^%. 

11  65f\%.       12  $56.       13  6%.       14  $672  left     $11.76,  int. 

71  Four  of  the  most  important  kinds  of  business  papers  are 
presented  in  this  section,  and  the  teacher  should  in  as  practical 
and  attractive  way  as  possible  lead  the  pupils  to  be  familiar  with 
their  form  and  to  be  able  to  write  them  without  assistance.  Trans- 
actions which  are  most  likely  to  occur,  and  which  involve  the 
making  of  bills,  promissory  notes,  receipts,  and  personal  accounts, 
should  be  given  in  addition  to  those  here  given.     Printed  blanks 


136  GRADED    ARITHMETIC.  [VI.  72 

may  be  provided  for  the  pupils  to  fill  out.  The  pupils,  however, 
should  not  depend  upon  these  forms,  but  learn  to  write  them  with- 
out aids  of  any  kind.  It  will  be  noticed  that  two  forms  of  bills 
are  given,  either  form  being  allowable,  except  for  bills  for  services, 
which  should  be  in  the  form  given  on  page  72,  except  that  the 
word  ''For"  instead  of  ''To"  may  be  used  if  desired. 

73    Answers:   2  Balance  due,  $186.40.      3  Balance  due,  $2.68. 
73    Answers:   1  Amount  of  receipt,  $153.75.     4  Balance,  $9.89. 

75  Answers:  2  Balance  due  from  K.,  $1.91.  3  Balance  due 
to  H.,  $0.26. 

76  Answers:  1  Amount  due  Jan.  24, 1894,  $177.6.3.  4  Amount 
of  bill,  $340.55  Cash  payment,  $169.81.  5  Balance  due  Parker, 
$5.02  Balance  due  Dexter,  $1.39  Balance  due  the  merchant, 
$2.59. 

Explain  in  4  the  custom  of  giving  credit,  that  is,  of  giving  a  cer- 
tain time  in  which  payment  may  be  made.  Also  explain  the  custom 
of  making  a  discount  Avhen  money  is  paid  before  it  is  due. 

It  would  be  interesting  for  members  of  the  class  to  open  an 
imaginary  account  with  each  other,  using  slips  of  paper  for  money, 
and  making  out  the  necessary  papers  that  pass  between  them. 

77  Answers:  1  $7.00.  2  $76.50.  3  34f  doz.  4  138iiyd. 
5  3  da.  6  Hi  da.  7  $2.70.  8  1079  pesos.  9  $669.  10  Mobile, 
92304000  cu.  in.  399584 -F  gal. ;  St.  Paul,  40032000  cu.  in.  173299- 
gal.;  Boston,  66816000  cu.  in.  289247-  gal.;  Chicago,  46368000 
cu.  in.;  200727+  gal.       11  27^  lb. 

Several  of  these  problems  should  be  performed  orally.  6  will 
doubtless  cause  some  difficulty.  Ask  such  questions  as  the  follow- 
ing if  necessary:  "At  the  end  of  three  days  how  many  days  would 
it  take  9  men  to  finish  it?  Suppose  all  had  left  but  one  man, 
how  many  days  would  it  have  taken  to  finish  the  work?  If  it 
takes  one  man  so  long,  how  long  would  it  take  4  men  ? "  Give 
similar  exercises  varying  the  conditions. 


VI.  78]  teachers'  manual.  137 

78  Answers:  1  19+  da.  $19.96.  2  43  bbl.  166^^  lb.  3  $153. 
4  $324.63.  5  S140.63.  6  $27.34.  7  $43.20.  8  i  of  a  pint. 
9  $54,085.  10  26y2^bags.  11  £16  8s  (9^0-  12  $3430. 
13  119  T.  1531  lb.  14  $21.78.  15  $141.86.  16  194^|  ft. 
17   8  ft.  wide. 

The  6  mo.  in  13  is  to  be  regarded  as  182^  days.  17  is  a  prac- 
tical problem  in  finding  the  greatest  common  divisor. 

79  A7isu-e)-s:  1  6  A.  74  rd.  93.5  sq.  ft.  2  |i.  3  $1.52 
$18.18f.  4f|^cd.  $5,188  +  .  5  67|fgal.  562^1b.  6  S18.487f 
8  50.2656  sq.  rd.  .31416  A.  9  12/^  cu.  ft.  10  250  lb.  11  1333^ 
lb.  12  1130.976  cu.  ft.  13  21120  ^  211.2"  21120^X  14  $41.60 
$78.40. 

In  7,  counting  Sundays,  the  answers  would  be  :  40033200  copies, 
and  $800064 ;  not  counting  Sundays,  the  answers  would  be : 
34329840  copies,  and  $686596.80. 

80  Ansicers:  1  $1402.50.  2  $37.50  per  mo.  3  $571.10. 
4  $1.74.  5  63  bu.  6  $198.45.  7  1§|  mi.,  or  1  mi.  223  rd.  |-ft. 
8  $17.  9  138JV  bu.  10  840  lb.  11  $320  $9120.  12  405 
steps.       13  ll^fif  f-  Ha.     381,Wu-       1*  $8.802|.       15  $1152. 

81  A7isu-ers:  1  63.1504+  bu.  2  51.130+  bu.  3  88  furrows 
511  A.  4  $12  less.  5  $2398.50.  6  84000.  7  27yV^. 
8  133.67+  oz.  33.41+  oz.  16.7+  oz.  9  $10.  10  8800.  11  $.02 
80%.  12  7i  m.  past  2  p.m.  13  41  yd.  15  in.  14  2.581  m. 
15  $36666|. 

The  answers  to  8  are  found  on  the  supposition  that  1  cu.  ft.  of 
water  weighs  1000  oz.  Let  the  pupils  perform  the  problem,  reck- 
oning 1  lb.  of  water  as  measuring  27.7  cu.  in.  This  is  a  more 
accurate  measurement. 

82  Ansu-ers:  1  $21600.  2  $28.95.  3  $.12  per  gal. 
4  $15.39.  5  10000.  6  3200  lb.  7  5333^  lb.  8  36if  cu.  ft. 
0^^^  T.  9  124.2  T.  10  131|f  panes  132  posts. 
11  20106.24  sq.  rd.     8293.824  ft.       12  863.       13  $1080. 


138  GKADED   ARITHMETIC.  [VI.  83 

83  Ansivers:  1  75/.  2  100%.  3  $133^  26f.  5  300 
sq.  ft.  6  $16000.  7  560.2+  rev.  8  9^\  6O/5V  9  118  qt. 
boxes.  10  80  rails  4224  rails.  11  173  pills.  12  $1  per 
crate.  13  $24000.  14  $33.68.  15  $358.06+  6tVW4V5*V% 
$53369557.71. 

Physicians'  prescriptions  are  frequently  made  in  the  manner 
indicated  in  11.     The  given  quantity  should  be  read  8  drams  and 

2  scruples. 

84  A7isivers:  1  $10440.80.  2  969i  bu.  3  $1.80.  4  Mon- 
day Wednesday.  5  $1.15  gain.  6  $3.38§f.  7  $1620. 
8  10890  sq.  ft.       121  ft.         9  $8,121.         10  800  da.       333^  da. 

11  Cheaper  by  the  dozen  $1  cheaper.  12  13fi|J  A. 
13  $4142  gain. 

85  Answers  :  1  864  books.  2  6  widths  30  yd.  7  widths 
37^  yd.         3  Apr.  10,  1876.         6  15  lb.       1  gal.  1  gi.         7  3  oz. 

6  oz.  8  493i  ft.  8/7  mi.  9  8  lots.  10  80  doses.  11  88 
bottles.  12  1411  yd.  13  $2.19.  14  205  pills.  15  22.79  bbl. 
16  86.78+  bu. 

86  Ansivers :  1  $672.  2  $480.  3  1215  cu.  ft.  4  243^  yd. 
5  77^1  A.      6  48  rd.      7  $163.74+.      8  13||  cd.      9  27  b.  20  b. 

10  12  da.  Ida.  11  136  men.  12  l^\)u.  13^.  14  $30000 
$64000.       15  $3.46.       16  $1309^^- 

87  Ansivers:   1  19712  bricks    14601+.     2  316|  rev.    34^^™!- 

3  396  tiles.  4  i%  of  5000.  6  5^^%.  7  Chicago  ahead 
211%.       8  $2812.50.       9  IIUI/-      10  ^O^a  yd.      11  $14.91|. 

12  $68.40.  13  1/0  tla.  14  $30000  D  $2373^  E.  15  960 
steps.       16  $4118.40. 

88  Answers:    2  i^^  (whole,  3184).  3  lyW^  ft.        '^i^S  ^^^ 

4  50166869     62633336     42.09%.       6  About  ^\    About  2\  times. 

7  389+  times.        8  4.38%     9.05%.        9  11.65%.       10  16.22%. 

11  28.16  gal  12  420^5.  13  2017888§  bbl.  14  189177083^ 
bu. 


VI.  89]  teachers'  manual.  139 

89  The  answers  to  these  exercises  will  be  approximate  only, 
and  will  therefore  vary  greatly.  As  the  purpose  of  the  work  is 
more  to  get  the  pupils  to  measure  on  the  map  and  to  learn  valuable 
information  in  geography  than  to  acquire  arithmetical  knowledge 
and  skill,  close,  accurate  work  should  not  be  insisted  upon.  Let 
the  pupils  measure  by  the  aid  of  the  scale  of  miles  to  which  the 
map  is  drawn,  and  also  by  the  aid  of  parallels  and  meridians. 

90  A7iswers:  1  32G2S.1824  lb.  13111.9452  1b.  2.23998. 
3  294148.008  lb.  374520  lb.  4  588060  lb.  5  30000  lb. 
6  3  oz.  17  pwt.  12  gr.  7  34  ft.  8  339.87"'.  9  1699.35'"  or 
337.89257  rd.  10  1  mi.  237.52+  rd.  11  37.88  sec. 
12  202.73  rds. 


SECTION   IX. 

NOTES   FOR   BOOK   NUMBER   SEVEN. 

"For  general  suggestions  as  to  the  kind  of  work  peculiar  to 
advanced  grades  in  Arithmetic,  teachers  are  referred  to  the  "  Note 
to  Teachers"  on  pages  iv  and  v,  particularly  to  what  is  said  of 
rules,  definitions,  and  explanations  of  problems.  In  these  forms 
of  expression  special  attention  should  be  given  to  accuracy  of 
statement,  and  by  accuracy  is  meant  not  merely  a  close  adherence 
to  any  one  particular  form,  but  an  exact  expression  of  a  logical 
process.  Loose  statements  which  earlier  in  the  course  were  allowed 
to  pass  should  now  be  challenged  and  corrected.  The  pupils  should 
be  led  by  judicious  questioning  to  detect  inaccuracies  of  their  own 
as  well  as  of  others'  statements. 

A  rule  in  Arithmetic  is  a  general  statement  of  the  method  of 
solving  a  given  class  of  problems.  A  rule  should  therefore  be 
derived  from  the  process  of  solution,  and,  as  far  as  possible,  be 
made  by  the  pupils  themselves.  The  rule  thus  made  by  the  pupils 
may  be  amended  by  means  of  criticism  and  comparison  with  a 
correct  form. 


140  GRADED   ARITHMETIC.  [Vll. 

A  definition  is  a  general  statement  of  the  peculiar  and  charac- 
teristic properties  of  the  subject  defined.  An  analysis  of  the  subject 
to  be  defined  should  be  made,  and  from  the  analysis  the  definition 
should  be  deduced,  if  possible,  by  the  pupils,  with  such  correction 
as  they  may  be  able  to  make.  For  example,  if  it  is  desired  to 
teach  the  definition  of  addition,  the  teacher  should  lead  the  pupils 
to  perform  or  to  recall  the  process  of  addition,  and  in  that  process 
to  see  that  they  have  found  a  number  which  contains  as  many 
units  as  all  the  given  numbers  taken  together.  After  being  told 
that  this  process  is  addition,  the  pupils  will  say  :  "  Addition  is  the 
process  of  finding  a  number  which  contains  as  many  units  as  all 
the  given  numbers  taken  together."  On  being  told  that  the  number 
found  is  called  the  sum,  they  will  correct  the  definition  given,  and 
say  :  "  Addition  is  the  process  of  finding  the  sum  of  two  or  more 
numbers,"  If  this  definition  is  still  thought  to  be  incomplete, 
attention  may  be  called  by  examples  to  the  fact  tliat  only  numbers 
of  the  same  kind  can  be  added,  as,  8  pounds  and  6  pounds.  8 
pounds  and  6  ounces  might  be  put  together  to  express  a  denominate 
number,  as  8  lb.  6  oz.,  but  such  putting  together  is  not  addition. 
The  pupils  will  therefore  further  correct  their  definition,  and  say: 
"Addition  is  the  process  of  finding  the  sum  of  two  or  more  numbers 
of  the  same  kind." 

An  explanation  of  a  problem  should  be  made  in  such  a  way  as 
to  be  easily  understood,  and  be  so  comprehensive  as  to  include  a 
statement  of  the  process  of  solving  the  problem  and  reasons  for 
each  step  of  the  process.  It  is  not  advised  that  all  or  many  of 
the  problems  performed  be  explained,  but  as  many  of  each  class 
of  problems  sliould  be  explained  as  to  give  assurance  that  they 
are  thoroughly  understood.  In  this  and  the  following  book  it  is 
recommended  that  the  processes  only  of  many  of  the  problems  be 
indicated,  either  orally  or  in  writing,  it  being  assumed  that  the 
mere  computations  may  be  accurately  ])erformed. 

In  the  work  of  previous  books  the  attention  of  teachers  has  been 
called  repeatedly  to  the  importance  of  illustrations  by  plans  or 
diagrams.     The    illustrations   should   be  continued  in  connection 


VII.  1]  teachers'  manual.  141 

with  all  difficult  problems  ;  but  for  the  solution  of  problems  not 
very  difficult  the  pupils  should  not  depend  upon  such  helps,  but 
imagine  and  state  the  conditions  as  nearly  as  possible.  If,  how- 
ever, their  statements  are  questioned  or  are  not  understood,  they 
should  be  ready  to  make  the  needed  illustrations. 

1-6  The  solution  of  these  review  problems  will  give  the  pupils 
little  difficulty,  if  the  previous  work  has  been  well  done.  When- 
ever difficulty  is  found,  lead  the  pupils  by  questions  to  see  exactly 
what  is  required,  and  what  steps  are  necessary  in  the  solution.  For 
example,  if  the  pupils  find  difficulty  in  performing  7,  page  3,  ask 
the  questions:  ''Before  you  can  find  the  worth  of  the  remainder, 
what  must  you  find  ?  How  can  that  be  done  ? "  If  the  pupils 
choose  to  find  in  a  roundabout  way  the  number  of  gallons  remain- 
ing, let  them  so  find  it ;  but  after  it  is  found  ask  them  if  there 
is  not  a  shorter  way.  Some  bright  pupil  will  see  that  26  qt.  =  6-^ 
gal.,  and  that  this  added  to  13^  gal.  is  19f  gal.,  which  subtracted 
from  28^  gal.  gives  8f  gal.  In  such  exercises  as  8,  page  3,  let  the 
pupils  make  a  diagram  before  the  solution  is  made. 

7  A7iswers:  1  /^  ^V%  ?\  tVVtt  Ml  ^h-  2  $32  $184.73 
$5.94  $140.04.  3  $52.50.  4  2  rd.  3  yd.  5  76  yd.  8  in. 
6  155rd.  l^ft.  $927.61  $1902.27.  7  10  yd.  8  80  rd.  120  rd. 
200  rd.  288  rd.  9. 11875  mi.  .16098+ mi.  10  $1760.  11  $389.78 
$406.74.  12  120  cups.  13  22  bu.  3  pk.  14  tIt  tVt  ^i 
^3^2^7^     ||5.       15  $50     $112.50     $9.15     $185.38. 

In  the  solution  of  11,  lead  the  pupils  to  find  the  number  of 
cubic  inches  or  feet  that  a  cental  or  hundredweight  occupies.  One 
bushel,  or  60  lb.,  occupies  2150.4  cu.  in.;  100  lb.  would  occupy  Yts' 
of  2150.4,  or  3584  lb.  The  solution  of  the  first  part  of  the  exer- 
cise would  be  indicated  thus  : 

<37  X  19  X  6  X  1728\ 


$1.15  X  (' 


2   X  3   X      3584 


8  Answers :  1  201  A.  20  sq.  rd.  2  11  A.  55  sq.  rd.  181^  sq.  ft. 
3  5  A.  17642.'^+ sq.ft.  4  837^^:^  ft.  5  50.26+ sq.  ft.  6  12.7  + A. 
7  $1360.     8  926.25 ^     9  $15200.     10  (a)  $524.38;  (b)  $2855.25; 


142  GRADED    ARITHMETIC.  [VII.  9 

(c)  159 posts;  ((I)  $57.04;  (e)  $31.07;  (/)  1027^^  eu. yd.  11  36 
ft.  12  5624.64  oz.  13  5j%%  cu.  ft.  1296  cu.  ft.  14  llJ  cu. 
yd.     l^^VA  cu.  yd.       15  3^%%  cords.       16  41  ft.  2*f  in. 

9  Answers:  1  (a)  519  sq.  ft.;  (b)  177i  sq.ft.;  (c)  19f  i  sq.  yd. 
(<;)  $16.54;  (e)  38-1  sq.  ft. ;  (/)  across  the  room,  widtliwise.  2  99-i-J- 
cu.  ft.  4  6  ft.  5  6J|  cords.  6  1615f  f  gal.  180  sq.  ft. 
7  $560     $120.       8  18150  cu.  ft.     $2637.42.       9  $264. 

10  Answers:  1  4800.  2  35.2+ qt.  3  115f^|cu.  ft.  4  $2.69 
$1.35  $.68  $7.35  $24.  5  $3  $3.06  $2.76  $2.45.  6  130f 
loaves  45.9+%.  7  166f  %  $.093|.  8  $14.24.  9  828.8  T. 
417.2  T.  10  $5180.  11  405  bu.  12  3if  i.  13  $40500. 
14  T%%     Ui     $44.60. 

11  Answers:   1  Amt.  of  bill,  $383.20.       2  $6120.       3  12^^;^. 

4  46|%  lost.  5  12038-  rd.  6  1  h.  6  min.  9  sec.  7  $418 
$33,589+.  8  4|%.  9  73^  planks.  10  $19,125  $.149+. 
11 220/^  yd.     $326.11.       12  $1743.09. 

12  Ansivers:   1  If ff  da.      2  $4876.      3  5*  mo.      4  $18,375. 

5  2.75.  6  $14375.  7  25.43+  sec.  8  33J§  T."  $67.86.  9  567| 
cu.yd.  10  22ff%.  11  180  mi.  12  529.21b.  13  $64  $14.80. 
14  $1,125     $3.60     $2.30.       15  $44.84. 

13  Answers:  6  28|/.  7  9|/.  8  $15.64  $21.16.  9  $69.12 
$453.12. 

Before  giving  these  exercises  in  Profit  and  Loss,  it  might  be 
well  to  tell  the  pupils  something  of  the  common  practice  of  mer- 
chants in  receiving  a  profit  for  the  trouble  of  buying  and  selling 
goods,  and  to  give  a  few  examples. 

14  Answers:  5  33^%  33^%.  6  11^%.  7  llJ%  on  sugar 
16f%  on  kerosene.       8  77|%.       10  22|%.       11  U^\%. 

Lead  the  pupils  to  see  that  the  per  cent  of  gain  or  loss  depends 
not  only  upon  the  amount  of  gain  or  loss,  but  also  upon  the  cost. 
This  is  illustrated  in  5.  In  9,  the  supposition  is  that  the  part 
unsold  is  worth  ^  of  what  was  paid  for  the  farm.     In  this  case  the 


vn.  15]  teachers'  manual.  143 

answer  would  be  25%.  Ask  the  question  :  "  What  was  the  per  cent 
of  gain  on  the  part  sold  (33^%)?"  If  necessary,  lead  the  pupils 
to  use  numbers  as  $2000,  for  the  cost  of  the  farm.  The  cost  of 
the  part  sold  would  then  be  f  1500  ;  and  if  it  were  sold  for  $2000, 
the  gain  Avould  be  33^%.  Again  ask:  "What  would  be  the  per  cent 
of  gain  if  I  should  sell  the  other  fourth  at  the  same  rate  ?  " 

After  the  pupils  are  familiar  with  the  process,  the  common  frac- 
tional part  of  the  cost  may  be  omitted,  the  per  cent  of  cost  being 
found  directly. 

15  A7iswe)-s:  4  ICf/  3^/.  5  |40.  6  $2.50.  7  $3.75.. 
8    $90.  9    $5         $5.75.  10    $4f.  11    Lost  $26f., 

12  $16304.35.       13  $1.074§. 

In  explaining  these  problems,  the  pupils  should  begin  with  the^ 
statement  that  the  cost  is  called  for,  and  that  it  equals  i§^  of  itself.. 
The  rest  of  the  explanation  may  be  given  as  indicated  in  the 
written  analyses,  the  reason  for  each  step  being  added ;  thus,  in 
3  :  '( $360  =  selling  price  =  |-§§  of  cost,  ^-i^  of  cost  =  ^i^  of 
$360  or  $3.  $3  =  t1o  of  cost,  laa  of  cost  =  100  times  $3  or 
$300." 

After  the  pupils  are  familiar  with  the  solution  of  this  kind  of 
problems,  the  use  of  hundredths  as  given  in  the  analysis  may  be 
omitted,  and  simpler  fractions  may  be  substituted ;  thus,  in  11 
the  solution  might  be  : 

$200  =  125%  or  -|  of  cost  of  1st  horse, 
f  of  $200  =  $160  =  cost  of  1st  horse. 

$200  =  75%  or  f  of  cost  of  2d  horse. 


of  $200  =  $266f  =  cost  of  2d  horse. 


$66f ,  loss  on  2d  horse  —  $40  gain  on  1st  horse  =  $26f  =  net  loss. 

Problems  may  also  be  performed  by  a  short  process  on  a  line ; 
thus,  in  12  : 

$10000  XlOO  X3  ,    n    -u . 
— —      ^^  =  cost  of  ship. 

Pupils  should  always  be  able  to  explain  problems  performed  in 
such  a  way.     The  explanation  of  the  above  solution  would  be,  "  If 


144  GRADED   ARITIOIETIC.  [VII.  16 

the  loss  was  8%,  the  selling  price  would  be  92%  of  the  cost. 
$10000,  the  selling  price  =  92%  of  the  cost  of  f  of  the  vessel. 
1%=^!^  of  $10000,  found  by  dividing  by  92,  and  100%  =  100 
times  this  quotient.  $10000  -^  92  X  100  =  cost  of  f  of  the  vessel. 
■^  of  the  vessel  cost  ^  of  this  sum,  found  by  dividing  by  2,  and  f , 
or  the  cost  of  the  vessel,  is  3  times  this  quotient." 

16  Answers:  6  25%  gain.  7  86^/.  8  30/.  9  72 baskets. 
10  $3     $2.72t\.       11  48§|%  gain. 

Lead  the  pupils  to  define  these  terms  in  the  manner  indicated 
on  page  140  of  the  Manual.  The  following  definitions  may  assist 
teachers  in  securing  concise  and  comprehensive  statements  : 

The  base  in  Profit  and  Loss  is  the  number  of  which  the  per  cent 
of  gain  or  loss  is  to  be  found.     It  is  usually  the  cost. 

The  rate  per  cent  is  the  number  of  hundredths  that  is  taken  of 
the  base. 

The  percentage  is  the  number  found  by  taking  a  certain  per  cent 
of  the  base.  The  percentage  in  Profit  and  Loss  is  the  amount  of 
gain  or  loss. 

The  amount  is  the  number  which  includes  the  base  and  percent- 
age, and  is  the  selling  price  in  Profit  and  Loss  when  the  percentage 
is  the  amount  of  gain. 

Tlie  remainder  is  the  difference  between  the  base  and  percentage, 
and  is  the  selling  price  in  Profit  and  Loss  when  the  percentage  is 
the  amount  of  loss. 

The  rules  called  for  in  3  to  5  should  be  deduced  by  the  pupils 
from  the  corresponding  processes,  and  may  be  as  follows  : 

To  find  the  gain  or  loss,  multiply  the  cost  by  the  rate  per  cent. 

To  find  the  rate  per  cent,  divide  the  gain  or  loss  by  the  cost, 
carrjnng  the  division  to  hundredths. 

To  find  the  cost,  divide  the  selling  price  by  100,  increased  by 
the  rate  per  cent  of  gain,  or  decreased  by  the  rate  per  cent  of  loss, 
and  multiply  by  100. 

17  Ans^vers:  1  $11095.20  154^\,%.  2  G8^^%  40f%  5^^%. 
3  45«3%  loss.       4  $1820     30%. 


VII.  18]  TEACHERS*   MANUAL.  145 

Some  of  the  exercises  from  5  to  20  will  have  two  sets  of  answers 
according  as  the  percentage  is  gain  or  loss.  Let  the  pupils  make 
these  problems,  having  the  percentage  either  profit  or  loss  as  they 
choose. 

18  The  explanation  included  in  1  may  have  to  he  enlarged  or 
extended  by  the  teacher.  Possibly  other  examples  than  these  given 
will  assist  the  pupils  to  understand  more  fully  the  business  of 
commission.  Besides  giving  the  names  of  the  consignor  and  con- 
signee, the  pupils  should  be  led  to  answer  in  general  that  <'the 
Consignor  is  the  person  who  sends  the  goods  to  another,"  and  "  the 
Consignee  is  the  person  to  whom  the  goods  are  sent." 

The  person  for  whom  a  buyer  or  seller  of  goods  acts  is  the 
PrincijKil. 

The  person  who  is  authorized  to  buy  or  sell  goods  for  another  is 
called  an  agent. 

The  allowance  made  to  an  agent  for  buying  and  selling  goods  is 
the  Commission. 

Attention  should  be  called  to  the  fact  that  the  base  in  com- 
mission is  the  cost  or  selling  price,  according  as  the  agent  is  a 
buyer  or  seller. 

19  Amwers:  3  1840.  4  $2500.00.  5  13947.33.  6  $49.07. 
7  21%.  8  $2125.  9  $5625.20.  10  $27.60.  11  $880. 
12  10279.50. 

The  pupils  should  be  able  to  tell  after  reading  over  a  problem 
whether  the  base  is  given  or  not.  Custom  varies  somewhat  as  to 
Avhat  the  base  should  be  for  estimating  the  commission  in  case  of 
a  selling  agent  who  pays  expenses  of  cartage,  storage,  etc.  When 
not  otherwise  specified  the  selling  price  is  regarded  as  the  base. 

^30  Answers:  4  $45.  5  $75  $2075  (if  premium  is  counted). 
6  $1500.       9  $79.50.       10  $2637.50  more.       11  $47.93. 

Bring  to  the  class  insurance  policies  of  various  kinds.  Blanks 
of  policies  may  be  procured  from  insurance  agents,  or  policies 
owned  by  the  parents  of  pupils  may  be  borrowed.  Explain  all 
the  terms  that  are  used  in  the  policies,  and  the  purpose  and  plan 


146  GRADED    ARITHMETIC.  [VII.  21 

of  carrying  on  the  business  of  insurance.  The  methods  of  organ- 
izing a  joint  stock  company  and  some  features  of  such  a  company 
are  shown  on  pages  49  and  50,  Book  VII.  Teach,  as  previously 
shown,  the  following  definitions  : 

A  iwlky  is  a  written  contract,  insuring  security  against  loss  be- 
tween two  parties. 

The  'premium  is  a  sum  paid  for  insurance. 

The  undenvriter  is  the  insurer. 

Refer  to  the  fact  that  mutual  companies  usually  charge  a  higher 
rate  of  insurance  than  stock  companies,  but  that  the  excess  is  off- 
set by  dividends  declared.  Give  problems  involving  insurance  in 
both  kinds  of  companies  to  ascertain  which  would  involve  the  least 
cost,  taking  into  account  the  interest  of  the  excess  paid  to  a  mutual 
company. 

21    Answers:   1  f%.  2  $1936.  3  $40.  4  $241.13. 

5    $2666|.         6    $2842.        7   $30000.        8   2%.  9  $1000. 

10  $2500.       11  $23985.       12  $1675.       13  $5  less. 

23  Answers:  3  1^%  $.0125.  4  $88.80.  5  $12  $.012 
$40.80.       6  1tW5%     $74.56.       8  $16,912     $718.76. 

In  the  preliminary  teaching  lesson,  use  familiar  examples,  both 
in  showing  the  purpose  of  taxation  and  the  meaning  of  the  various 
terms  used.  From  recent  reports  of  assessors,  give  actual  facts  of 
the  valuation  and  rate  of  taxation  of  the  city  or  town  in  which 
the  pupils  live. 

Real  estate  is  fixed  property,  such  as  lands  and  houses. 

Personal  properttj  is  property  that  is  movable,  such  as  money, 
stocks,  cattle,  furniture,  etc. 

A  poll  tax  is  a  tax  upon  the  person  of  a  citizen. 

Assessors  are  officers  whose  business  it  is  to  estimate  the  value 
of  property,  and  to  apportion  among  individuals  the  sum  to  be 
raised. 

In  finding  the  method  of  assessing  a  tax,  get  from  the  local 
assessors  the  method  tliat  is  actually  employed  by  them.  In  some 
places  the  assessors  are  paid  a  regular  salary,  and  in  assessing  the 


VII.  23]  teachers'  manual.  147 

tax  their  salaries  are  not  taken  into  account.  The  amount  of  tax 
uncollectible  cannot  be  known  before  the  tax  is  collected,  but 
frequently  allowance  is  made,  based  upon  the  experience  of  past 
years.  The  following  is  a  practical  rule  for  finding  the  rate  of 
taxation  where  the  overlay  is  not  more  than  5*^  of  the  sum  to  be 
raised  on  the  property: 

Divide  the  net  sum  to  be  raised  by  96,  and  multiply  the  quotient 
by  100.  From  this  sum  subtract  the  poll  tax,  and  divide  the  differ- 
ence by  the  amount  of  taxable  property.  If  the  overlay  is  greater 
than  5^,  the  first  divisor  must  be  correspondingly  less. 

23    Answers:    2  $52.50.      3  $53.40.      4  $134.10.      5  $18.75. 

6  $98.70.       7  $212763.72.       8  $4500. 

In  7,  deduct  the  uncollectible  tax  before  finding  the  cost  of 
collecting  or  net  proceeds. 

34  Ansivers:   3  $188.31.      4  $46.19.       5  $100.       6  $656.64. 

7  $338.62.       8  $145.       9  $42. 

In  teaching  this  subject  show  the  reasons  which  governments 
have  for  taxing  imported  goods.  The  following  facts  might  be 
presented  in  one  form  or  another  : 

In  addition  to  the  usual  duties  collected  at  the  custom-house,  a 
tax  called  tonnage  is  levied  upon  vessels  according  to  the  number 
of  tons  they  can  carry.  In  some  places  duties  are  paid  upon 
articles  of  export.  An  invoice  of  goods,  called  also  a  manifest, 
sometimes  gives  a  wrong  valuation.  To  guard  against  this  fraud, 
a  certificate  from  the  consul  is  sometimes  required,  stating  that  the 
prices  given  in  the  invoice  are  the  prevailing  market  prices.  The 
terms  leakage  and  breakage  are  used  to  denote  the  allowance  for 
waste  of  liquids  and  for  the  breaking  of  bottles.  Unless  otherwise 
specified,  the  value  of  a  pound  sterling  in  U.  S.  money  is  reckoned 
at  $4,866. 

35  Ansivers:  1  $888.  2  $17.53.  3  $18.19.  4  $2881.25. 
5  $97.18.  6  $1687.50.  7  $2.13.  8  $252.76.  9  300  lb, 
10  $45.68. 


148  GRADED   ARITHMETIC.  [VII.  26 

36  Anstvers:  1  $40.70  |2679.30.  2  4325  bu.  3^451.50. 
4  $405292.50  $371518.12|-.  5  $60.  6  $500  loss.  7  $2209.80. 
8  $1,008.       9  1%.       10  $12466.02.       11  $6155.85. 

In  8,  if  2240  lb.  are  taken  as  a  ton  the  answer  is  90/. 

21    Answers:   1  $326.40  loss.  2  $3,323.  3  $.003473 

$46.65.        4  $29.62.        5  $2866.20.        6  $838.50.        7  $659.50. 

8  1%     3%     1%         9  $1900.59.         10  18  machines. 

28    Answers:   1    U%.  2    $1680.22.  3    Silk  @  $5.57 

Ribbon  @  $.295       Lace  @  $1.03.  4  $5866f.  5  $644.32. 

6  $18625.  7  li%.  8  $853.13.  9  A,  $20.51  B,  $23.67 
C,  $42.60     D,  $12.62.       10  $695.76.       11  $5585.08. 

39  These  problems  should  be  performed  orally  and  by  the 
shortest  way.  Where  the  way  is  not  indicated  in  the  question,  let 
the  pupils  choose  the  method  of  solution. 

30   Answers:   4  .142.     5  .091.     6  .1815.     7  .036^.     8  .0715. 

9  .097.  10  .041|.  11  .063.  12  .036f.  13  .135^.  14  .062^. 
15  .037^.  16  .211|.  17  .2461.  18  .093^.  19  .279. 
20  .127f.  21  .2285.  22  .362|.  23  .178,1.  24  .109|. 
25  $18.67.  26  $43.05.  27  $33.37.  28  $38.15.  29  $180.49. 
30  $23.75.  31  $24.08.  32  $109.12.  33  $37.62.  34  $20.05. 
38  $411.25.   39  $278.85. 

While  occasionally  the  method  of  casting  interest  is  determined 
by  peculiar  conditions,  it  is  generally  best  to  adopt  and  follow  one 
method.  A  method  very  generally  pursued  by  business  men  is 
what  is  called  the  60-day  method.  By  this  method  the  interest  is 
ascertained  first  for  60  days  by  getting  -j^i^  of  the  principal,  and 
for  other  times  from  this  sum.  For  example,  if  it  is  required,  as 
in  25,  to  find  the  interest  of  $200  for  1  yr.  6  mo.  20  da.  at  6%, 
first  divide  by  100  to  find  the  interest  for  2  mo.,  and  multiply  that 
sum  by  9  to  find  the  interest  for  18  mo.  20  da.  is  ^  of  60  da.,  and 
therefore  divide  the  interest  for  2  mo.  by  3.  Add  results  for  the 
final  result.     The  solution  will  appear  as  follows  : 


VII.  30]  teachers'    ISIANUAL.  149 

$200.00  Principal. 

2.00  Int.  for    2  mo. 
18.00     "      "    18  mo.  (2  mo.  X  9) 
.67     "     "    20  da.  (^  of  60  da.) 


$18.67     "     "    18  mo.  20  da. 

Not  all  problems  can  be  so  easily  performed  by  this  method,  but 
a  little  practice  will  give  the  pupils  skill  in  finding  the  desired 
fractional  parts.  Thus,  in  30,  find  the  interest  for  2  mo.,  then  for 
15  da.  by  dividing  the  interest  for  2  mo.  by  4,  then  for  2  da.  by 
dividing  the  interest  for  00  da.  by  30  (^  of  ^i^).  In  33,  multiply 
tlie  interest  for  2  mo.  by  2  to  find  the  interest  for  10  mo.  Divide 
the  interest  for  2  mo.  by  2  to  find  the  interest  for  1  mo.  Get  ^^^ 
of  the  interest  for  1  mo.  to  find  the  interest  for  27  da.  This 
solution  will  appear  as  follows  : 

Interest  of  $632.29    for  11  mo.  27  da.? 
6.322  Int.  for  2  mo. 
$31.61    Int.  for  10  mo.  (2  mo.  X  5). 
.316  3.16      "      "      1  mo.  (^  of  2  mo.). 
9                           2.84      "      "    27  da.  (j%  of  30  da.). 


2.844  $37.61      "      "    11  mo.  27  da. 

When  this  method  is  familiar  to  the  pupils,  only  the  times  need 
be  written  after  each  partial  interest.  The  following  solutions, 
furnished  by  business  men,  illustrate  this  method  of  casting  interest 
actually  used  in  business  : 

Int.  of  $2907  from  Dec.  2,  '90, 

Int.  of  $4120  from  Aug.  26,      to  May  27,  '92,  @  5%. 
'90,  to  May  27,  '92,  @  5%.  17  mo.  25  da.  @6%  on  $2907. 

21  mo.  1  da.  @  6%  on  $4120. 
^  per  mo. 


20  mo. 

412. 

1  mo. 

20.60 

1  da. 

.68 

433.28 

>ss  ^ 

72.21 

$361.07 


16  mo. 

232.56 

1  mo. 

14.54 

20  da. 

9.69 

3  da. 

1.45 

2  da. 

.97 

259.21 

less  ^ 

43.20 

$216.01 


150  GKADED    AEITEBIETIC.  [^'"H*  31 

The  above  method  is  a  slight  modification  of  what  is  called  the 
.six  per  cent  method.     By  this  method  the  interest  of  $1  for  the 
•given   time    at    6^^    is    multiplied   by  the    number   of   dollars  in 
.the  principal,  and  if  the  rate  is  other  than  6%,  the  interest  found 
is  increased  or  diminished  as  required.     The  interest  of  $1  for 
1  yr.  at  6%  is  $.06  ;  for  2  mo.,  $.01 ;  for  1  mo.,  f.OOo  ;  for  6  da., 
$.001 ;  and  for  1  da.,  i  of  a  mill.     In  general,  the  interest  of  $1 
is  found  by  taking  six  times  as  many  cents  as  there  are  years,  one- 
half  as  many  cents  as  there  are  months,  and  one-sixth  as  many 
mills  as  there  are  days.     By  a  little  practice  one  is  able  to  give  the 
interest  of  $1  at  6%   for  any  time  very  quickly,  and  if  the  6% 
.method  is  pursued  it  will  be  good  practice  for  pupils  to  do  this. 

31  Answers:  1  $213.33.  2  $111.88.  3  $61.25.  4  $20.35. 
5  $29.89.  6  $66.72.  7  $46.39.  8  $8.15.  9  $19.70. 
10    $138.03.  11    $8.59.  12    $25.95.  13     $792.25. 

14  $868.49.  15  Apr.  10  and  Oct.  10  of  each  year  to  Oct.  10, 
1893,  each  payment  being  $43.75.  The  last  payment,  $1793.75, 
including  principal  and  six  months'  interest,  was  made  Oct.  10, 
1893.  16  $250.30.  17  $64  $.175+  $12.80  $25.95. 
18  $8.17.       19  $20.96.       20  $35.55.       21  $34.14. 

In  finding  the  time  between  two  dates  for  business  purposes, 
there  is  a  difference  of  practice  among  business  men.  A  common 
practice  in  estimating  time  for  casting  interest,  is  to  regard  each 
month  as  consisting  of  30  days  ;  thus,  in  finding  the  time  from 
Sept.  20,  1891,  to  Aug.  1,  1892,  it  is  11  months  to  Aug.  20,  and  to 
Aug.  1,  it  is  19  days  less  than  11  months,  or  10  mo.  11  da. 

At  the  time  of  purchase  (Ex.  15),  Mr.  Brown  received  from  Mr. 
Smith  a  deed  of  the  property,  and  at  the  same  time  gave  to 
Mr.  Smith  two  papers  signed  by  him,  one  a  note  promising  to  pay 
Mr.  Smith  $1750  with  interest  at  5^  payable  semi-annually,  and 
the  other  a  deed,  called  a  mortgage  deed,  providing  that  the 
property  shall  come  into  full  possession  of  Mr.  Smith  in  case  the 
terms  of  the  note  are  not  fulfilled.  This  deed  must  be  acknowl- 
edged before  a  Justice  of  the  Peace  or  Notary  Public. 


VII.  32] 


TEACHERS     MANUAL. 


151 


To  find  the  accurate  interest,  find  the  interest  for  the  exact 
number  of  days,  regarding  365  days  as  the  year.  The  following 
solution  of  18  will  illustrate  the  method  : 


30 
31 
30 
_1 
92 


rind  the  accurate  interest  of  $540  for  92  da.  at  0%. 
108 
^$M  X  .06  X  02 


■m 

73 


108 

.06 

6.48 

92 

12  96 

.     583  2 

73)596.16(8.166+ 
584_ 
121 
73 
486 
438 


8.17  A 


us. 


480 


=  .1f;8.166+ 


To  find  the  exact  number  of 
days  between  July  1  and  Oct. 
1,  lead  the  pupils  to  say  :  "  30 
more  days  in  July,  31  in 
August,  30  in  September,  and 
1  in  October.  Adding,  we 
have  92  days."  The  explana- 
tion of  the  problem  is  simply: 
"Multiply  $540  by  .06  to  get 
the  interest  for  1  year.  Divide 
this  product  by  365  to  find  the 
interest  for  1  day,  and  multi- 
ply by  92  to  find  the  interest 
for  92  days." 


32  Answers:  3  $15.30  4%  6.84+%.  4  6%  5%  3|-+%. 
5  4^%  5.611%.  6  4%.  7  13i+%.  8  4i%.  9  6.53+%. 
10  6%.  11  4^f%.  12  3.45+%.  13  3.41+%.  15  $210. 
5  yr.  16  1  yr.  2  mo.  12  da.  17  1  yr.  3  mo.  18  9  mo.  22  da. 
19  1  yr.  7  mo.  1  da.  20  5  mo.  9  da.  21  1  yr.  6  mo.  21  da. 

22    2  yr.  5  mo.  8  da.  23    4  mo.  24    3  yr.  3  mo.  17  da. 

25    2  mo.  17  da.  26    4  yr.  8  mo.  20  da.  27    4  mo.  4  da. 

28  3  yr.  11  mo.  9  da. 

DraAV  reasons  as  well  as  results  from  the  pupils,  leading  them  to 
say  in  explanation  of  1 :  "  Since  at  ojic  per  cent  the  interest  for 
the  given  time  is  $1  to  yield  an  interest  of  $20,  it  will  take  as 


152  GRADED   ARITHMETIC.  [VII.  33 

many  per  cent  as  $1  is  contained  times  in  $20,  whicli  is  20. 
Answer,  20^."  The  form  of  explanation  may  vary  somewhat ;  for 
example,  the  reason  for  dividing  may  be  expressed  :  "  To  yield  $20 
interest  the  rate  must  be  as  many  times  one  per  cent  as  there  are 
times  $1  in  $20,  which  is  20." 

The  form  of  written  solution  of  this  class  of  problems  may  be  as 
follows  : 

6   Find  the  rate  per  cent. 

$480  X  .01  X  3  =  $2.40  int.  at  1%. 
$9.60 -^  $2.40  =  4. 
Answer,  Aofo. 

As  in  finding  the  rate  per  cent  the  interest  at  one  per  cent  is 
made  the  divisor,  so  in  finding  the  time  the  interest  for  one  year  is 
used  in  a  similar  way.  The  explanation  of  this  class  of  problems 
may  be  as  follows  :  "  Since  in  one  year  $400  Avill  gain  at  the  given 
rate  $20,  it  will  take  as  many  years  for  it  to  gain  $60  as  $20  is 
contained  times  in  $60,  which  is  3.  Answer,  3  years."  The 
written  solution  of  17  would  appear  as  follows  : 

$800  X  .05  =  $40  int.  for  1  yr. 
$50  -^  $40  =  li. 

Aiiswer,  1  yr.  3  mo. 

The  common  custom  is  to  drop  the  fraction  of  a  day  if  it  is  less 
than  ^,  and  to  add  a  day  if  it  is  ^  or  more  than  ^. 

33  Answers:  3  $266.67.  4  $150.  5  $810.  6  $1425. 
7  $2306.49.  8  $1370.30.  9  $3378.64.  11  $4.  12  $16. 
13  $620.  14  $1021.504+.  15  Int.,  $75.95  ;  Amt.,  $915.95. 
16  Time,  6  mo.  12  da.;  Amt.,  $412.80.  17  Kate,  4|%  ;  Amt., 
$350.60.  18  Prin.,  $713,958  +  .  19  Time,  7  mo.  27  da.; 
Amt.,  $8700.  20  Rate,  4.39+ %  ;    Amt,  $1700.  21  Prin., 

$360.54+  ;  Amt.,  $387.04  +  .     22  Int.,  $13,848  ;  Amt.,  $294,248+. 
23  Time,  9  yr.  2  mo.  10  da.;  Amt.,  $1480.40. 

Essentially  the  same  principle  is  involved  in  the  solution  of 
these  problems  as  in  the  solution  of  problems  in  which  the  rate  or 


VII.  34]  teachers'  :manual.  158 

the  time  is  sought.  A  principal  of  one  dollar  is  taken  when  it  is 
desired  to  find  the  principal.  The  following  written  solutions  of 
6  and  12  will  illustrate  a  good  method  : 

Find  the  principal. 

II  X  .01  -^  2  =  $.005  int.  of  $1  for    1  mo. 
$.005  X  9  =  $.045    ''     "    ''     '-      9  mo. 
$.005  -^  5  X  3  =:  $.003    "     "    ''     "    18  da. 
$68.40  4- $.048  =  $1425. 

Answer,  $1425,  principal. 

Find  the  principal. 

$1  X  .04^  X  2|-  =  $.ll^  int.  of  $1  for  2  yr.  6  mo.  at 

$1.1125)  $17.8000  (16 
11125 


6  6750 

6  6750 

Answer,  . 

^16,  principal. 

34    Answers :   1 

5  yr.       2i 

yr.       3  yr.  1  mo.  15 

da. 

2  6% 

3 

1  yr.  8  mo.       4 

16|  yr.     25 

yr.     14f  yr. 

5  59^ 

1- 

6  $36000 

7 

H%- 

8  Jan. 

1,  1897. 

9  $652.63. 

10 

May  20,  1894 

11  July  1, 

1891. 

12  $2500. 

13  $14400 

$21600. 

14  4% 

15  $375  loss. 

In  15,  by  "  average  expense  "  is  meant  average  yearly  expense, 

35  Ansxvers:  3  $100.  4  $100  cash.  5  $200.  6  394.088+. 
7  $345,394.       8  $177,639.       9  $1578.947. 

No  new  principle  is  involved  in  these  problems.  The  present 
worth  is  the  principal,  which,  put  at  interest,  will  amount  to  a 
given  sum  in  the  given  time  and  rate.  The  explanation  draAvn 
from  the  pupils  may  be  (3):  "The  amount  of  $1  for  the  given  rate 
and  time  is  $1.05.  It  will  require  as  many  dollars  to  amount  to 
$105  as  $1.05  is  contained  in  $105,  which  is  100.    Answer:  $100." 

36  Answers:  1  $294.12.  2  $21.06.  3  $123.53.  4  The 
latter  offer,  $41.14  better.  5  Less  than  true  value.  G  '3796 
$.02  $808.  7  $657.64.  8  $1188.12  Less  $1211.88+. 
9  $588.       10  $22.50.       11  $450.       12  $745.29. 


154  GRADED   AEITHMETIC.  [VII.  36 

Let  the  pupils  see,  in  such  problems  as  3,  that  the  answer  given 
is  the  present  gain,  the  present  worth  of  the  note  given  being 
regarded  as  the  actual  cost  in  cash. 

In  finding  the  true  value  of  the  note  (6)  March  1,  the  principal 
is  found,  which,  put  at  interest,  would  amount  to  $800  in  1  mo. 
at  6^.  Sometimes  more  than  one  discount  is  made  from  the  list 
price.  Show  this  to  the  pupils  by  giving  an  instance,  as,  for 
exainple  :  The  list  price  of  tacks  is  $16  per  cwt.,  the  cash  price 
being  at  a  discount  of  25%  and  10%  of  the  list  price,  or,  as  the 
seller  might  say,  25  and  10  off.  To  find  the  cash  price,  first  deduct 
25%  from  the  list  price,  and  then  10%  of  this  result.  Thus,  25% 
offl6  =  $4;  116  — 14  =  $12;  10%  of  $12  =  $1.20;  $12  — $1.20  = 
$10.80,  cash  price.  It  might  be  well  to  give  several  problems  of 
this  kind  to  the  pupils. 

37  Anstvers:  1  $53.04,  due  Sept.  1.  2  $44,  due  Jan.  1, 1894. 
3  $218.85. 

Pupils  should  be  led  to  write  out  fully  and  clearly  an  analysis 
of  each  problem,  and  to  give  an  explanation  of  each  item.  The 
following  written  solution  of  2  may  suggest  a  good  form  for 
teachers  to  present  as  the  given  questions  are  answered  by  the 
pupils  : 

Principal,  Jan.  1,  1893,  $100.00 

Interest  to  IVIay  1  (4  mo.),  2.00 

Amount,  102.00 

Payment  May  1,  40.00 

New  i)rincipal,  62.00 

Interest  to  Aug.  1  (3  mo.),  .93 

Amount,  62.93 

Payment  Aug.  1,  20.00 

New  principal,  42.93 

Interest  to  Jan.  1,  '94  (5  mo.),  1.07 

Amount  due,  Jan.  1,  '94,  $44.00 

38  Answers:  1  $254.44.  2  $112.81.  3  $255.30.  4  $88.24+. 
5  $132.20.     6  $258.42+.     7  $164.29+. 


Til.  38]  teachers'    ISIANtTAL.  155 

The  method  above  shown  of  finding  the  balance  due  on  a  note 
"when  partial  payments  have  been  made,  is  quite  generally  followed 
by  business  houses.  It  is  in  the  main  tlie  method  adopted  by  the 
United  States  Supreme  Court,  and  is  therefore  called  the  United 
States  rule.  One  feature  of  this  rule  provides  that,  "if  any  pa}^- 
ment  be  less  than  the  interest,  the  surplus  of  interest  must  not 
be  taken  to  augment  the  principal,  but  interest  continues  on  the 
former  principal  until  the  period  when  the  payments,  taken  to- 
gether, exceed  the  interest  due,  and  then  the  surplus  is  to  be  applied 
toward  discharging  the  principal."  Tell  this  to  the  jKipils,  and 
show  how  it  would  be  applied  in  7  if  the  payment  made  Sept.  25 
had  been  $10.  Show  how  it  would  be  applied  if  the  September 
and  January  payments  had  been  $10  and  $6  respectively.  Do  the 
same  with  other  exercises. 

Wlien  partial  payments  are  made  on  accounts  and  on  notes 
running  a  short  period  of  time,  the  interest  is  sometimes  computed 
by  the  following  ru.le  :  Find  the  amount  of  the  principal  from  the 
time  it  begins  to  draw  interest  to  time  of  settlement.  From  this 
amount  subtract  the  sum  of  the  payments  and  their  interests  from 
the  time  of  payment  to  the  time  of  settlement,  and -the  difference 
will  be  the  balance  due.  This  rule  is  sometimes  called  the  mer- 
chants' rule.  The  following  solution  of  5  is  by  this  rule  : 
Principal,  on  interest  from  April  1,  '92,  $300.00 
Interest  to  March  1,  '93  (11  mo.),  16.50 

Amount, 
Payment  June  1,  '92, 

Interest  to  March  1,  '93  (9  mo.). 
Payment  Oct.  1,  92, 

Interest  to  March  1,  '93  (5  mo.). 
Payment  Dec.  1,  '92, 

Interest  to  March  1,  '93  (3  mo.). 
Payment  Jan.  1,  '93, 

Interest  to  March  1,  '93  (2  mo.). 
Sum  of  payments  and  interests. 

Balance  due  March  1,  '93,  $131.90 


$316.50 

$50.00 

2.25 

60.00 

1.50 

30.00 

.45 

40.00 

.40 

184.60 

156  GRADED   ARITHMETIC.  [VII.  39 

Ask  the  pupils  to  perform  6  and  7  by  the  above  method,  and 
compare  answers  with  the  answers  obtained  by  the  United  States 
rule. 

39  Answers:  2  $195.06+  $197.49  +  .  3  $1427.33. 
4  $747.34+.       5  $248.94  +  .       6  Simple  int.  $30.14  greater. 

In  1,  draw  from  the  pupils  the  fact  that  compound  interest  is 
interest  on  interest.  Savings  banks  generally  allow  compound 
interest  on  deposits.  In  3,  interest  is  calculated  for  five  separate 
times,  the  last  being  8  mo. 

The  answers  to  question  calling  for  blanks  to  be  supplied  are  as 
follows:  1  yr.  @  4%,  1.04;  1  yr.  @  5%,  1.05;  3  yr.  @  3%, 
1.092727;  3  yr.  @  5%,  1.157625  ;  4  yr.  @  4%,  1.169859  ;  5  yr.  @ 
3%,  1.159274;  5  yr.  @  5%,  1.276282;  6  yr.  @  3%,  1.194052; 
6  yr.  @  4%,  1.265319;  7  yr.  @  3%,  1.229874;  7  yr.  @  5%, 
1.4071;  8  yr.  @  4%,  1.368569;  8  yr.  @  5%,  1.477455;  9  yr.  @ 
3%,  1.304773  ;  10  yr.  @  5%,  1.628895. 

40  Answers:  2  $1458.  3  $166.46.  4  $636  $710.16  $101 
$730.20.       5  $1786.20     $1257.64.       6  $2668.73. 

In  the  instance  given  in  1,  the  account  stands  as  follows,  if  no 
interest  is  paid  until  maturity  of  note  : 

Principal,  $100.00 

Total  simple  interest  (3  yr.  @  6%),  18.00 

Interest  on  1st  annual  interest  (2  yr.),  $0.72 

"  2d        "           "        (1  yr.),  0.36          1.08 

Amount  due  at  the  end  of  3d  year,  $119.08 

The  amount  by  compound  interest  for  the  same  time  and  rate  is 
$119.10,  only  2  cents  more  than  the  amount  by  annual  interest. 
In  4,  by  the  process  indicated,  $609.16  would  be  due  Sept.  1, 1886. 
If  the  note  were  paid  at  this  time,  $101  should  be  deducted  from 
$609.16.  The  simplest  solution  of  this  problem  is  to  find  the 
amount  due  as  if  no  payment  had  been  made,  and  then  subtract 
$100  plus  the  interest  of  $100  from  July  1, 1886,  to  time  of  settle- 
m«nt.     5  should  be  performed  in  the  same  way.     Let  the  pupils 


VII.  41]  TEACHEKS'    MANUAL.  167 

perform  6  in  two  ways,  one  by  the  United  States  method,  and  the 
other  as  if  the  interest  at  6%,  payable  annually,  was  unpaid  until 
maturity  of  note.  The  answer  given  above  is  obtained  by  the  latter 
method. 

41  Answers:  1  $9000.  2  1  mo.  1  da.  12  yr.  6  mo.  3  $100 
gain.  4  3  yr.  11  mo.  1  da.         2  yr.  11  mo.  8  da.  5  80%. 

6  $545.85.  7  3  yr.  5  mo.  22  da.  8  Cash  offer,  $45.45  cheaper. 
9  $64800.       10  $1243.92.       11  First  by  2/^%.       12  $2790.62. 

13  $1115.075     $1104.50     $1115.36. 

42  Answers:  1  Tlie  second.  2  $13333^  $35555f 
3   $1949.85.             4   5i%.             5   $15238^^-  6   $444.05. 

7  $5214.285.  8  $13.50  9  16/  18/.  10  22-|  yr. 
lllfyr.            11  $1459.44.           12   $480000.            13   $887.97. 

14  $307.80. 

43  National  Banks,  before  issuing  any  bills  of  their  own,  are 
obliged  to  })ut  a  certain  portion  of  their  capital  into  government 
bonds,  which  are  held  in  trust  for  the  banks  by  the  Treasury 
Department  at  Washington.  The  bills  "  payable  to  bearer  "  "  on 
demand  "  are  printed  at  Washington,  and  cannot  exceed  in  amount 
90%  of  the  face  value  of  the  bank's  deposit  in  the  Treasury 
Department.  The  stockholders  of  a  bank  are  its  owners,  and  may 
share  all  profits  and  losses  as  partners  in  proportion  to  the  number 
of  shares  they  hold.  If  a  bank  lessens  its  capital,  or  is  unable 
from  any  cause  to  pay  its  debts,  the  stockholders  are  obliged  to 
make  up  the  deficiency  not  exceeding  the  par  value  of  the  shares 
which  each  stockholder  owns.  The  books  and  papers  of  National 
banks  are  periodically  inspected  by  bank  examiners  appointed  by 
the  government.  The  board  of  directors,  besides  electing  the 
president,  cashier,  and,  if  needed,  other  officials,  are  supposed  to 
attend  to  the  general  management  of  the  bank. 

The  income  of  a  bank  is  derived  from  the  amount  received  in 
discounting  notes,  in  collection  of  notes,  in  the  interest  on  govern- 
ment bonds  and  other  securities,  and  in  profits  of  buying  and 
selling  property.     The  expenses  are  the  taxes  it  has  to  pay  to  the 


158  GRADED   ARITHMETIC.  [VII.  44 

general  government  and  to  the  state  and  local  governments,  and 
the  running  expenses,  such  as  rents,  salaries,  etc. 

The  above  facts,  and  others  which  may  be  gained  from  bank 
authorities,  should  be  given  to  the  pupils. 

44:  The  method  of  borrowing  money  at  a  bank  should  be  shown 
in  a  familiar  and  practical  way,  the  example  here  given  being 
taken  as  a  basis.  By  questioning  and  telling,  the  following  facts 
should  be  brought  out :  It  is  supposed  that  the  payee  goes  to  the 
bank  to-day  to  get  the  note  discounted.  He  has  first  to  indorse 
the  note,  that  is,  to  write  his  name  on  the  back  of  it.  By  this 
indorsement  he  guarantees  the  payment  of  the  note.  If  the  officials 
of  the  bank  require  greater  security,  another  person  must  indorse 
the  note.  The  payee  receives  from  the  bank  in  cash  $6.15  less 
than  $300.  This  is  called  the  avails,  and  is  really  less  than  the 
true  value  of  the  note.  If,  instead  of  presenting  the  note  to-day, 
the  payee  should  present  it  one  month  later,  the  bank  would  have 
to  wait  but  2  months  before  receiving  $300.  It  therefore  discounts 
the  interest  of  $300  for  2  months  and  3  days.  At  the  end  of  3 
months,  or  within  3  days  after  maturity  of  the  note,  Mr.  Douglas 
is  supposed  to  pay  $300  to  the  bank.  If  lie  does  not  do  this  after 
due  notice,  a  notice  called  a  "  protest "  is  sent  by  a  notary  public 
to  the  payee,  and,  if  he  does  not  pay,  tlien  to  the  other  indorser. 
The  indorser  must  pay  the  amount  at  once,  so  that  the  bank 
may  not  lose,  but  he  can  liold  the  maker  responsible  for  payment 
afterwards. 

Blanks  illustrating  the  business  of  a  bank  can  be  easily  obtained 
and  shown  to  the  pupils.  The  officials  will  be  willing  to  give  all 
needed  information. 

The  following  definitions  may  be  evolved  from  actual  examples 
given  : 

The  avaih  or  j^^oceeds  of  a  note  is  the  sum  that  the  bank  pays 
for  it  after  deducting  the  interest  on  the  face  of  the  note  for  the 
time  before  maturity. 

The  face  of  the  note  is  the  sum  of  money  written  in  the  body  of 
the  note. 


Vll.  45]  teachers'    ]MANITAL.  159 

The  maturity  of  the  note  is  the  time  at  which  the  note  is  due. 

Bank  Discount  is  the  allowance  made  to  a  1)ank  for  tlie  payment 
of  a  note  before  it  becomes  due. 

The  niakevoi  the  note  is  the  person  who  signs  his  name  to  the  note. 

The  2Hii/ee  is  the  person  to  whom  payment  of  money  is  promised. 

The  indorse)'  is  the  person  who  writes  his  name  on  the  back  of 
the  note. 

Days  of  grace  are  days  after  a  note  is  payable  before  it  is  legally 
due.  If  the  last  day  of  grace  falls  on  Sunday  or  on  a  legal  holi- 
day, the  note  must  be  paid  on  the  day  previous.  In  the  state  of 
New  York  no  grace  on  notes,  etc.,  is  allowed  by  law. 

Practice  varies  in  estimating  the  time  of  maturity  and  period  of 
discount  of  notes.  Some  banks  in  estimating  the  time  of  maturity 
of  a  note  due  in  90  days  regard  the  time  as  three  calendar  months, 
but  more  commonly  such  time  is  taken  in  exact  days.  In  discount- 
ing notes  some  banks  always  reckon  the  time  before  maturity  in 
exact  number  of  days,  others  in  months  and  days,  and  still  others 
in  exact  number  of  days  if  the  luiexpired  time  is  60  days  or  less, 
and  in  months  and  days  if  for  more  than  60  daj's.  The  problems 
of  this  section  are  reckoned  on  the  basis  of  months  if  the  time 
given  in  the  note  is  months,  and  of  exact  number  of  days  if  the 
time  given  is  in  days.  For  example,  the  maturity  of  a  note  dated 
July  15,  due  in  3  months,  is  Oct.  15  -|-  3  days  of  grace,  or  Oct.  18. 
If  the  note  should  read  "90  days  after  date,"  the  note  would  fall 
due  Oct.  l.">  +  3  days,  or  Oct.  16.  If  the  first  of  these  notes  is 
discounted  at  date  of  note  or  any  time  subsequently,  the  time  of 
discount  is  reckoned  as  months  and  days.  The  discount  of  the 
second  note  is  reckoned  in  exact  number  of  days. 

45  .l».s-?m-.s-;  1  $493.87.  2^590.70.  3  $594.75.  4  $787.60. 
5  $444.75.  6  $296.85.  7  $670.65.  8  $1267.63.  9  $182.48. 
10  $343.67  $346.53.  11  $674.05.  12  Nov.  15, 1893  $628.01 
$620.44     $612.53.         13  $452.92     $455.21. 

4.Q  Answers :  1  ^100  $1200.  2  $750.  3  $1000.  4  $510.46. 
5  $1755.91.  6  $561.59  $28.57.  7  $1012.42.  8  $8.20 
$591.80.       9  $607.85     $7.85.       10  $2000     $1979.       11  $6.35 


160  GRADED    ARITHMETIC.  [VII.  47 

$1253.65.      12  «5482     $344.82.     13  $600     $597,525.     14  $5.51 
$420.79.     15  $15.97. 

The  explanation  of  such  problems  as  1  may  be  :  "  If  a  note  for 
one  dollar  will  give  proceeds  of  $.9895,  it  will  take  a  note  for  as 
many  dollars  to  give  proceeds  of  $98.95  as  $.9895  is  contained 
times  in  $98.95,  which  is  100.     A7isiver:  $100." 

47  The  deposit  slip,  with  the  bank  book,  is  passed  to  the  teller, 
who  writes  in  the  bank  book  the  amount  of  deposit.  The  check 
by  which  $80  is  drawn  out  is  signed  by  the  person  who  wants  the 
money,  and  is  made  payable  to  himself  or  to  his  order.  The  check 
given  to  James  Smith  may  pass  through  various  hands  until  finally 
it  comes  to  the  Emporia  Bank.  It  would  be  well  for  the  teacher 
at  this  point  to  explain  the  bank  check  and  to  show  its  usefulness 
to  business  men  and  others.  To  do  this  it  will  be  necessary  to 
show  the  necessity  of  first  making  a  deposit  of  money  at  the  bank, 
and  to  tell  exactly  hoAv  it  may  be  used  in  paying  persons  who  live 
at  a  distance  as  well  as  persons  near  the  bank  upon  which  the  check 
is  drawn.  An  example  may  be  given  of  James  Robinson  living  in 
Albany,  and  wishing  to  pay  a  creditor  in  Omaha.  If  he  has  a 
deposit  in  a  bank  in  Albany,  he  has  only  to  fill  out  a  blank  similar 
to  the  blank  check  represented  on  this  page  and  send  it  to  the 
creditor  in  Omaha,  who  carries  it  to  a  bank  there  for  payment.  If 
he  is  known,  he  will  receive  the  face  value  of  the  check ;  if  he  is 
not  known,  he  must  be  "identified"  by  some  one  before  he  can  get 
the  check  "cashed."  The  banks  all  have  dealings  with  one  another, 
and  therefore  tlie  clieck  is  taken  by  the  Omaha  bank  and  returned 
to  Albany.  If  j\Ir.  Robinson  has  no  deposit  in  a  bank,  he  may 
make  a  special  deposit  of  a  sum  equal  in  amount  to  the  face  value 
of  the  check  which  he  wishes  to  send  to  Omaha,  or  he  may  for 
a  small  amount  buy  a  "cashier's  check"  for  the  desired  amount, 
payable  to  the  order  of  Mr.  Robinson  or  to  his  creditor.  If  to 
the  former,  Mr.  Robinson  writes,  "Pay  to  the  order  of  John  Brown," 
and  signs  his  name.  Find  out  from  bankers  the  exact  working  of 
a  "Clearing  House,"  and  explain  the  process  to  the  pupils. 

48  Answers:   1  $683.41.       2  $803.08. 


VII.  49-50]  teachers'  manual.  161 

Cooperative  banks  have  been  very  extensively  established,  and  are 
likely  to  be  more  common  in  future.  For  this  reason,  and  because 
it  will  be  useful  for  people  to  know  their  purpose  and  methods  of 
operation,  they  should  be  the  subject  of  study  in  school.  Their 
nature  and  methods  vary  so  much  in  different  parts  of  the  country 
that  it  has  not  been  thought  best  to  introduce  in  the  text-book 
more  than  a  statement  of  their  general  features.  It  would  be  well 
to  get  from  the  officers  of  the  nearest  cooperative  bank  its  method 
of  operation  and  give  to  the  pupils  problems  whicli  are  likely  to 
come  to  patrons  of  tlie  bank,  both  as  lender  and  as  borrower. 
Describe  fully  its  advantages,  and  make  the  problems  as  practical 
as  possible. 

49-50  In  naming  the  different  kinds  of  corporations,  only 
those  that  are  best  known  need  be  given,  such  as,  banks,  railroads, 
and  large  manufactories.  It  would  be  well  while  discussing  these 
subjects  to  show  the  pupils  the  various  documents  that  are  used, 
such  as,  chartei-,  bond  with  coupons,  and  bond  certificate  ;  also 
certain  blanks  which  banks  and  other  business  houses  have  to  fill 
out.  Lead  the  pupils  to  see  clearly  the  difference  between  bonds 
and  stocks,  bondholders  and  stockholders,  interest  and  dividend. 
Answers  to  the  most  difficult  questions  given  on  these  pages,  and 
some  others  that  ought  to  be  answered,  are  contained  in  the  fol- 
lowing brief  statement : 

A  corporation  is  an  organization  which  has  a  charter  to  do 
business  under  the  laws  of  the  state.  Its  capital  is  divided  into 
shares  owned  by  the  stockholders,  each  of  wliom  holds  a  certificate 
of  stock  stating  the  number  of  shares  owned  by  that  person.  The 
stockholders  elect  a  board  of  directors,  who  attend  to  the  general 
management  of  the  corporation.  The  par  value  of  tlie  shares  is 
the  original  or  face  value.  If  for  any  reason  the  shares  are  sold 
for  less  than  their  par  value,  they  are  said  to  sell  below  par,  or 
at  a  discount ;  if  for  more  than  their  par  value,  they  are  said  to 
sell  above  par,  or  at  a  premium.  The  profits  of  the  business  are 
divided  among  the  shareholders  in  proportion  to  the  number  of 
shares  that  each  one  holds.     These  profits  are  called  a  dividend. 


162  GRADED    ARITHMETIC.  [Vll.  51 

If  for  any  reason  a  corporation  desires  to  increase  its  capital,  it 
can  make  assessments  on  the  stockholders,  that  is,  oblige  them 
to  pay  a  certain  amount  for  each  share  owned,  or  it  can  borrow 
money  and  issue  promissory  notes  called  bonds,  which  are  given 
to  persons  as  security  for  the  money  they  lend.  They  are  made 
payable  at  a  certain  time  or  at  regular  times,  and  bear  a  specified 
rate  of  interest.  Attached  to  each  bond  are  printed  slips  stating 
that  so  much  interest  will  be  paid  the  bearer  at  a  certain  time. 
These  slips  are  called  coujwns,  and  can  be  cut  off  as  they  are 
needed.  If  the  business  of  the  corporation  is  good,  and  the  in- 
terest promised  is  large,  the  bonds  are  likely  to  sell  above  par,  or 
above  their  face  value.  If  the  interest  promised  is  very  low,  or 
if  there  is  danger  of  not  pa^dng  the  interest,  the  bonds  are  likely 
to  sell  below  par,  or  at  a  discount. 

A  stock-broker  is  a  person  who  buys  and  sells  stocks  and  bonds. 
If  this  is  done  for  other  people,  he  charges  a  commission  called 
brokerage.  The  brokerage  is  generally  estimated  upon  the  par 
value  of  the  stocks  or  bonds. 

51  Ansicers:  1  $1080  $60.  2  $2210  $100.  3  25  shares 
$4762.50  4/^\%.  4  $4400  $300  (S^j%.  5  $60  on  a  thousand 
5.28%.  6  4  80.  7  8.  8  Stock  *§%  better.  9  600  shares. 
10  118375.  117^%  6i%  4|%  5f%.  12  Stock  1%  better 
Stocks  lif§%  better.       13  $18987.50. 

52  Ansivers:  1  80  120  133^  75  70.  2  4%.  3  6%. 
4  25%.  6  $1672.50.  7  $5400.  8  $8010.  9  $5835. 
10  $26193.75.  11  $10003.13.  12  $3069.50  $15347.50. 
13    7%  bonds.       14  Bonds  about  1%  better.       15  $2(100. 

53  Explain  to  the  pupils  the  great  convenience  of  the  use  of 
drafts  or  checks,  and  how  drafts  are  exchanged  in  different  parts 
of  the  country.  Show  that  business  firms  and  private  individuals 
in  one  part  of  the  country  owe  parties  in  another  part,  and  that  by 
a  system  of  exchange  these  debts  may  be  paid  without  sending  the 
money.  Illustrate  this  by  the  example  indicated  on  this  page. 
Mr.  Macomber,  who  lives  in  Chicago,  owes  John  Smith  of  Boston, 


VII.  54]  teachers'  manual.  163 

$480.90.  He  goes  to  Mr.  Upliam,  a  banker  in  Chicago,  and  buys 
of  him  an  order,  called  a  draft,  on  Thos.  S.  Appleton  &  Co., 
bankers  of  Boston,  who  have  an  account  with  Mr.  U})ham.  Mr. 
j\[aconiber  writes  on  the  back  of  the  draft,  "  Pay  to  the  order  of 
John  Smith,"  and  signs  his  name.  This  he  sends  to  Mr.  Smith, 
who  draws  the  money  from  Appleton  &  Co.  at  the  given  time.  If 
the  draft  had  read  ten  days  after  sight,  it  should  first  be  presented 
to  Appleton  &  Co.  for  acceptance,  which  is  done  by  their  writing 
the  word  '* accepted"  and  signing  the  firm's  name  across  the  face 
of  the  note.  Ten  days,  +  3  days  of  grace,  after  acceptance  the 
money  is  payable.  This  is  called  a  "time  draft,"  to  distinguish 
it  from  a  "  sight  draft,"  which  is  payable  "  at  sight." 

If  in  two  different  parts  of  the  country,  as,  for  example,  Boston 
and  San  Francisco,  the  amount  of  drafts  drawn  in  each  city  on 
the  other  is  nearly  equal,  the  buyer  of  a  draft  in  either  city  will 
have  to  pay  little  or  nothing  beyond  the  face  value.  But  if  the 
amount  of  drafts  in  Boston  on  San  Francisco  is  far  in  excess  of 
the  amount  drawn  in  San  Francisco  on  Boston,  then  the  buyer  of  a 
draft  in  Boston  will  have  to  pay  a  premium  to  pay  for  the  risk 
and  expense  of  sending  money  to  San  Francisco,  and  the  buyer  of 
a  draft  in  San  Francisco  may  be  able  to  get  it  at  a  discount. 

54  A7iswers:  1  $1015  fl004..50.  2  f  1465.50.  3  $609. 
4  $831.60.  5  $531.06.  6  $732.17.  7  $879.30.  8  $762.02. 
9  $900.10.  10  $804.42.  11  $609.14.  12  $5940.27. 

13  $8074.83.       14  $1962.17.       15  5iio%. 

In  1,  the  time  draft  costs  less  than  the  sight  draft,  because  the 
banker  has  the  use  of  the  money  for  63  days,  including  3  days  of 
grace. 

$1015  — $10.50,  the  interest  of  $1000  for  63  days  =  $1004.50. 

Another  method  of  solution  is  to  find  the  cost  of  $1  of  exchange 
and  multiply  by  $1000,  thus  : 

$1  +  $.015  =  $1 .015  $1,015  -  $.0105  =  $1.0045 

$1.0045  X  1000  =  $1004.50. 


164  GRADED    ARITHMETIC.  [VII.  55 

Another  solution  of  9  may  be  as  follo^\'s  :  $1  of  exchange  can  be 
bought  for  $1.01.  As  many  dollars  of  exchange  can  be  bought  for 
$1000  as  $1.01  is  contained  times  in  $1000.  $1000  ^$1.01 
=  990.10.     Ansiver,  $990.10. 

The  time  draft  mentioned  in  9  would  cost  .9945  of  the  face. 
$1000 -^  .9945  =  face. 

The  solution  of  10  is  as  follows  :  The  cost  of  $1  of  exchange  is 
found  by  subtracting  $.0155,  the  interest  on  $1  for  93  days,  from 
1000  =  $9845.  To  this  add  $.01,  the  premium  =  $.9945.  Since 
one  dollar  of  exchange  can  be  bought  for  $.9945,  as  many  dollars 
of  exchange  can  be  bought  for  $800  as  there  are  times  $.9945  in 
$800,  etc. 

55  Answers :    2  $488.25.       3  $334.62.       4  £367  145.  5d. 

Show  the  pupils  that  essentially  the  same  principle  is  involved 
in  foreign  exchange  that  has  already  been  explained  in  domestic 
exchange.  The  intrinsic  value  of  the  standard  coins  of  two 
countries  determines  the  par  of  exchange  between  those  countries. 

Exchange  is  at  a  premium  or  discount  when  the  market  value  is 
more  or  less  than  the  par  value.  Three  bills  of  the  "  same  tenor 
and  date "  are  sent  by  different  mails,  so  that  if  one  is  lost  the 
other  may  be  presented. 

In  the  bill  of  exchange  given,  Mr.  Mason  is  the  buyer  or  remitter 
who  indorses  the  bill  payable  to  the  order  of  Mr.  Brown.  As  in 
domestic  exchange,  foreign  bills  or  drafts  are  made  payable  both 
on  time  and  at  siglit.  Time  drafts  are  usually  quoted  at  a  less 
price  than  sight  drafts. 

56  Answers:  1  £587  15s.  2d.  2  $1679.69.  3  $1104. 
4  14392  francs.  5  $2367.47.  6  35  bonds,  $95.85  surplus. 
7  $24.50.  8  7%.  9  $11520.  10  $454.78.  11  $325.03. 
12  $84.50.       13  $2  more  for  buying  land.       14  $162.49. 

The  note  refcri-cd  to  in  11  is  supposed  to  bo  discounted  at  a  bank. 

51  Answers :  1  $2VJ.rA)  8%.  2  U\i%.  3  $787.60  6.09+%. 
4  7.15+%  0.09+%  9.21+%  4..55+%.  5  18.4+%.  6  $304.50. 
7  $1267.55.       8  $723.84.       9  $679.61.       10  $406.20    $402.21. 


VII.  58] 


TEACHERS     MANUAL. 


165 


58  It  will  not  be  necessary  for  the  pnpils  to  perform  many  of 
the  problems  included  in  1.     A  few  answers  are  given  below  : 

(1)  1500  Atch.,  f!1038.75.  (3)  2000  C.  B.  &  Q.,  $2250.  (8)  2000 
Or.  Sh.  Line  6's,  111811.33.  (16)  42  B.  &  M.,  f  5717.25.  (24)  5 
Conn.  Eiver,  )i;i075.63.  (26)  20  Mex.  Cen.,  $126.25.  (37)  235 
Bos.  &  Mont.,  $4141.88.     (46)  96  Pullman,  $13240. 

59  Answers:  1  Balance  March  31,  $58.27.  2  Balance  Jan.  31, 
$182.21. 

If  preferred,  the  two  sides  of  tlie  cash  account  may  be  placed  on 
the  opposite  pages  of  a  blank  book. 

60  A7isivers :   1  Balance,  $221.52.      3  Balance,  $1049.43. 

The  form  suggested  in  4  is  given  below.  It  has  the  advantage 
over  other  forms  in  the  ease  with  which  the  balance  is  found,  and 
also  in  the  greater  convenience  for  making  out  monthly  bills. 

L.  P.  WALKER. 


189Jf. 

Jan.  1 

To  Balance, 

$18 

60 

u      a 

By  Cash, 
To  Balance, 

10 

00 

8 

60 

«    s 

"  10  gal.  K.  Oil, 

11 

1 

10 

«     5 

"  2  Ih.  Coffee, 

32 

H 

te      i( 

"  3^  lb.  Cheese, 

12 

J^ 

"     9 

"  3  gal.  Molasses, 

4^ . 

1 

26 

((     « 

''  1  hhl.  Flour, 

5 

60 

17 

62 

((     (( 

Btj  Cash, 
To  Balance, 

15 

00 

2 

62 

(jl— 0'^  In  uuiking  the  cash  account,  use  the  form  already 
given.  The  other  accounts  may  be  given  in  the  simple  form  given 
above.  The  poultry  account  should  be  made  precisely  as  the  per- 
sonal accounts  are  made,  debiting  all  items  of  expense  incurred 


166  GEADED    ARITHMETIC.  [YII.  63 

on  account  of  poultry,  and  crediting  all  items  of  income  from  the 
poultry. 

63  A7iswers :  1  8  mo.  18  mo.  200  mo.  2  18  mo.  60  mo. 
800  mo.  3  In  2  mo.  11  da.  4  2  mo.  from  date.  5  82^  da. 
6  3  mo.  3  da.       7  Oct.  14.       8  2  mo.     1200  mo.     li  mo. 

Explain  to  the  pupils  that  the  average  of  payments  is  the  time 
at  which  several  items  of  debt  due  at  different  times  may  be  paid 
without  gain  or  loss  to  either  party. 

A  form  of  written  solution  for  6  is  as  follows  : 
2  mo.  X    420  =    840  mo. 

7Ut  da. 


3  mo.  X  315  -- 

4  mo.  X  525  : 

=  945  mo. 
=  2100  mo. 

365  X  30 
1260 

10950  , 
=  1260  ^"- 

1260 

)  2885  (2  mo. 
2520 

365  Jnsiver :  2  mo.  8  da. 

In  finding  the  equated  time,  when  in  the  result  the  fraction  of 
a  day  is  one  half  or  more,  reckon  it  one  day;  if  less  than  ^,  dis- 
regard it. 

In  9,  assuming  that  the  two  sums  owed  were  paid  at  the  time 
the  first  sum  was  due,  the  manufacturer  would  lose  the  use  of 
$1200  from  June  20  to  July  5,  or  15  days,  which  is  equivalent  to 
the  loss  of  the  use  of  $1  18000  days,  or  tlie  loss  of  the  use  of 
$2000  9  days.  Therefore,  for  the  manufacturer  not  to  lose  any- 
thing, the  $2000  should  be  kept  by  him  9  days  after  June  20,  or 
until  June  29. 

04    Answers :   1  June  16.       2  Sept.  26.       3  Feb.  11. 

When  each  of  two  parties  is  a  debtor,  one  way  of  finding  the 
equated  time  of  paying  the  balance  due  is  by  computing  the  in- 
terest. In  4,  it  might  be  assumed  that  the  account  given  is  the 
account  of  each  pupil  with  J.  li.  8i)aulding.  The  explanation  may 
be  drawn  from  the  puj^ils  by  such  questions  as  are  given  in  the 
exercise.  Afterwards  the  pupils  may  be  led  to  explain  the  problem 
as  follows  : 

'■'■Xn  this  account  Spaulding  owes  me  $100,  due  July  1,  and  .*!50, 
due  July  21.     I  owe  Spauldiug  $100^  due  July  11.     I  wish  to  find 


VII.    64:] 


TEACHERS     MANUAL. 


167 


the  time  when  he  may  pay  the  balance  ($50)  due  me  without  gain 
or  loss  to  either  of  us.  Wi  will  assume  July  21  as  the  time  of 
payment  of  all  the  sums.  If  he  pays  me  the  $100  due  July  1 
on  the  21st  of  July,  I  shall  lose  the  use  of  $100  from  July  1  to 
July  21,  or  20  days.  The  interest  of  $100  for  20  days  at  6%  is 
33-J/.  If  he  pays  me  July  21  $50,  which  is  due  July  21,  I  shall 
neither  gain  nor  lose.  If  I  pay  him  July  21  $100,  which  is  due 
July  11,  I  shall  gain  the  use  of  $100  10  days.  Tlie  interest  of 
$100  for  10  days  is  lOf /.  If  payment  of  the  balance  due  is  made 
July  21,  my  net  loss  is  16f /.  That  I  may  not  gain  or  lose,  he 
must  pay  me  the  $50  as  many  days  before  July  21  as  it  takes  $50 
to  gain  lOf/,  or  20  days.  20  days  before  July  21  is  July  1,  when 
the  balance  should  be  paid." 

Spaulding  owes  me  :  I  owe  Spaulding :  Assumed  time  of 

$100  due  July    1.  $100  due  July  11.         payment,  July  21. 

50    "    Jidy  21. 
Loss,  int.  of  $100  20  da.  =^--  .33^.        Gain,  int.  of  $100  10  da.  =  .16f . 

Net  loss,  .16|. 
Interest  of  $50  1  da.  =  .008^.  .16|  -^  .008^  =  20. 

July  1  —  20  da.  =  July  1. 

By  introducing  the  last  question  a  new  problem  is  made.  Assume 
that  the  last  bill  of  goods,  like  the  others,  is  due  in  1  mo.  Let 
Aug.  1  be  the  assumed  time  of  payment.  The  written  solution  by 
a  good  method  may  be  as  follows  : 


Due 

Amount 

Time 

Interest 
lost 

Due 

Amount 

Time 

Interest 
gained 

July    1 
July  21 
Aug.     1 

$100 

50 

100 

31  da. 

11  da. 

0 

$.516 
.091 

July  11 

$100 

21  da. 

$.35 

250 
100 

.607 
.35 

$150 

.257- 

■^  .0025  =  1 

0 

Aug.  1  — 10  daijs  =  Jul^  2U 


168  GRADED   ARITHMETIC.  [VII.  65 

Explanation  :  If  $100  due  me  July  1  is  paid  Aug.  1,  I  lose  the 
interest  of  $100  31  da.,  or  $.516.  If  $50  due  me  July  21  is 
paid  Aug.  1,  I  lose  the  interest  of  $50  11  da.,  or  $.091.  If  I 
pay,  July  11,  $100  that  is  due  Aug.  1,  I  gain  the  interest  of  $100 
21  da.,  or  $.35.  If  all  payments  are  made  Aug.  1,  my  net  loss  is 
$.257.  That  I  may  neither  gain  nor  lose,  he  should  pay  the  bal- 
ance ($150)  due  me  as  many  days  before  Aug.  1  as  the  number  of 
days  it  will  take  for  $100  to  gain  $.257.  Dividing  by  the  interest 
of  $150  for  1  day  the  quotient  is  10.  10  days  before  Aug.  1  is 
July  21,  the  equated  time. 

Many  problems  in  averaging  accounts  may  be  solved  best  by 
reducing  the  gain  or  loss  of  $1  for  a  number  of  days  or  months, 
and  then  dividing  by  the  balance  due.  For  example,  4  could  be 
solved  as  follows  : 

Spaulding  owes  me :  _  r,       i  t 

*100  due  July    1.  iT,'''T    '"f,^ 

50    "    July  21.  «100  due  July  U. 

If  settlement  is  made  July  21, 

I  lose  the  use  of  I  gain  the  use  of 

$100  for  20  da.  =  the  use  of  $1       $100  for  10  da.  =  the  use  of  $1 

2000  da.  1000  da. 

Net  loss  of  use  of    $1  for  1000  da. 

<<      "     "     "    "  $50   ''       20  da. 

July  21  —  20  da.  =  July  1. 

65  Armvers:  1  June  15.  2  Nov.  18.  3  Balance,  $260, 
due  April  1.       4  Oct.  27. 

These  problems  are  essentially  the  same  as  the  problem  solved 
above.  The  earliest  date  instead  of  the  latest  may  be  taken  as 
the  assumed  time  of  payment  if  thought  best. 

06  Answers :  1  June  4,  1893.  2  Balance  due  July  16,  1894. 
3  June  13,  1894.  4  March  2,  1895  $891.09.  5  May  19, 1879 
Balance  due,  $44.74. 

67  A  horizontal  line  is  a  line  having  the  direction  of  a  line 
in  the  surface  of  still  water.     A  vertical  line  is  one  that  has  the 


VII.  08]  teachers'  manual.  169 

direction  of  a  phimb  line.  These  lines  need  be  drawn  only  approxi- 
mately in  the  right  direction.  Parallel  lines  may  be  drawn  by  the 
aid  of  a  square  or  ruler,  care  being  taken  to  draw  them  so  that 
they  shall  be  the  same  distance  apart  throughout  their  length. 
Other  ways  of  drawing  parallel  lines  will  be  shown  later. 

A  line  may  be  divided  into  2  equal  parts  by  means  of  a  measure. 
Another  way  is  to  divide  it  with  the  aid 
of   compasses,  as  follows  :  With  A  as  a  '■■•l--''^ 

centre,  draAV  the  arc  CD,  and  with  Jj  as  a  /l\ 

centre,  draAV  the  arc  EF,  cutting  the  arc  /    ;    \ 

CD.     Unite  the  points  of  intersection  of        .  /I      • 

^  ...  .  .     .  d ;        'G     : B 

the  two  arcs,  and  the  dividing  line  divides  1     T      i  " 

the  line  AB  at  G  into  2  equal  parts.     Each  \     :     / 

of  the  two  parts  can  be  divided  in  the  same  \  !  / 

way    into    2    equal    parts,    thus   dividing  X 

the  original  line  into  4  equal  parts. 

68    Most  of  these  exercises  are  a  review  of  previous  work. 

09  The  conclusion  in  1,  after  several  similar  experiments,  is, 
that  the  sum  of  two  sides  of  a  triangle  is  greater  than  the  third 
side.  The  two  ways  of  iinding  by  experiment  the  sum  of  the  angles 
of  a  triangle  are  :  (1)  Cutting  off  the  corners  of  a  paper  triangle 
and  measuring,  and  (2)  Measuring  by  means  of  a  protractor. 

Several  conclusions  may  be  made  from  the  measurements  made 
in  17;  such  as  :  The  two  diagonals  of  a  i:»arallelogram  divide  each 
other  into  two  equal  parts.  In  a  square  the  diagonals  are  equal. 
In  a  square  the  diagonals  are  perpendicular  to  each  other.  In  a 
rhombus  the  diagonals  are  })erpendicular  to  each  other. 

70  Answers:  1  $80.  2  129600  sq.  ft.  3  HI  ^-  5  11^^^ 
sq.  rd.  6  T^V  ^^-  9  2244  sq.  ft.  10  21G00  sq.  ft.  11  1  T. 
1966  lb.       12  176.59+  ft.       13  85*  rd.      14  19  planks     $6.08. 

In  9,  the  Avalk  is  siipposed  to  be  on  the  border  of  the  garden. 

71  Answers:  1  23^  yd.  2  $3.30.  3  20^1^  sq.  ft.  4  869.5 
sq.  ft.  5  7855.8  sq.  ft.  $116.26.  6  $46.78  $3.92  (full  allow- 
ance for  openings)    $24.75   460^^^  sq.  ft.     8  118-^^^  sq.  rd.     9  140  ft. 


170  GRADED   ARITHMETIC.  [VII.  72 

For  a  good  practical  method  of  finding  the  number  of  rolls  of 
paper  required  for  a  room,  see  page  119  of  the  Manual. 

73  Answers:  1  20000  sq.  ft.  $3443.53.  6  2  A.  4  sq.  rd. 
7  1  A.  126.2+  sq.  rd.     11  435.6  ft. 

The  land  taken  by  the  railroad  company  in  1  is  a  parallelogram 
whose  base  is  ef  and  whose  altitude  is  equal  to  the  distance  bd. 

For  solution  of  2,  see  Book  IV.,  page  85. 

The  dotted  line  called  for  in  8  is  perpendicular  to  the  base,  since 
the  altitude  is  the  perpendicular  distance  from  the  vertex  to  the 
base.     Let  the  statements  be  as  concise  as  possible. 

'73  Answers:  1  5e\^s(i.  ixl.  1936  yd.  4.54.^^^^  £c.  4826|  sq. 
ft.  16  A.  3^  sq.  ch.  45  A.  8J|  sq.  ch.  15f  rd.  2  1200  sq.  ft. 
3  1212^  sq.  ft.  4  1732^  sq.  ft.  5  6383^  sq.  ft.  6  1758f 
sq.  ft.       7  900  sq.  ft.       8  920  sq.  rd. 

The  line  required  to  be  drawn  60  ft.  long  in  7  is  supposed  to  be 
horizontal. 

74  Answers:  2  8^  sq.  ft.  3  22400  sq.  ft.  46200  sq.  ft. 
5  2  inches.  6  Area  of  triangle,  2400  sq.  ft.;  of  trapezoid,  2100 
sq.  ft.        7  1450  sq.  ft. 

The  cut  on  page  12,  Book  VI.,  will  suggest  a  solution  for  1. 

4  may  be  solved  by  construction,  after  transforming  a  trapezoid 

7?  -4-  /) 
into  an  equivalent  rectangle  ;  also  from  the  formula  A  =  — - —  X  h. 

At 

75  Ansimrs:  1  6  sq.  ft.  2  12600  sq.  ft.  11  36.34  sq.  ft. 
14  25.1328  ft.  15  4.77+  ft.  16  314.16  sq.  ft.  17  9.58+  A. 
18  $4.36. 

Suggestive  helps  to  the  solution  of  these  exercises  will  be  found 
on  pages  14-18  of  Book  VI. 

Use  protractor  or  compasses  in  marking  off  arcs  of  various 
degrees. 

A  regular  polygon  is  a  polygon  having  equal  sides  and  equal 
angles.  In  drawing  regular  polygons,  draw  circles  first  and  mark 
off  equal  arcs  of  the  desired  length. 


VII.  76]  teachers'  manual.  171 

Let  the  pupils  divide  paper  circles  into  triangles  before  drawing 
if  they  have  not  already  done  so  (see  page  18,  Book  VI.). 

76  AnsH-ers:  1  502.656  sq.  ft.  3  22  ft.  138.2304  ft.  175.9296 
ft.     932.48  sq.  ft.       4  $41.30.       5  492  A.  145.4+  s(i.  rd. 

In  1,  it  is  supposed  that  the  cow  is  tethered  in  such  a  way  as  to 
enable  her  to  graze  at  a  distance  exactly  forty  feet  from  the  post. 
'  Somewhat  similar  work  with  prisms  is  called  for  on  page  19, 
Book  VI.  For  suggestions  in  teaching  these  forms,  see  Manual, 
page  120.  The  following  statements  may  be  drawn  from  the 
pupils  : 

The  lateral  faces  of  all  right  prisms  are  in  the  form  of  rectangles. 
The  bases  of  a  triangular  prism  are  triangles.  The  bases  of  quad- 
rangular prisms  are  quadrilaterals  ;  etc.  A  jJcntaffonal  j-jm-m  is  a 
prism  having  pentagons  for  its  bases.  A  rirjht  prism  is  a  prism 
whose  lateral  edges  are  perpendicular  to  the  bases.  An  oblique 
2))'is>ii  is  a  prism  whose  lateral  edges  are  inclined  to  the  bases. 

The  drawing  on  page  64,  Book  V.,  will  suggest  the  kind  of 
work  called  for  in  8. 

77  Answers:  1  245  cu.  ft.  2  1152  cu.  in.  3  57f||  bbl. 
8  20.5632  gal.      9  8  ft.  6.1241-  in.    21  ft.  9.4379  in.     10  79.5872 

sq.  ft. 

The  altitude  of  a  jjp'amid  is  the  perpendicular  distance  from  the 
vertex  to  the  base.  The  slant  height  of  a  reyular pyramid  is  the 
perpendicular  distance  from  the  vertex  to  any  side  of  the  base. 
(First  show  that  a  regular  pyramid  is  a  pyramid  which  has  a  regu- 
lar figure  for  the  base,  and  the  vertex  directly  above  the  center 
of  the  base.  All  pyramids  referred  to  in  this  book  are  regular 
pyramids.)  Teach  and  ask  the  pupils  to  describe  triangular,  quad- 
rangular, pentagonal,  hexagonal,  and  octagonal  pyramids. 

For  suggestions  in  answering  5  and  6,  see  page  20,  Book  VI. 

The  altitude  of  a  cone  is  tlie  perpendicular  distance  from  the 
vertex  to  the  base.  The  slant  height  of  a  cone  is  the  distance  from 
the  vertex  to  any  point  in  the  circumference  of  the  base.  (All 
cones  referred  to  in  this  book  are  right  circular  cones.     A  right 


172  GRADED    AKITHMETIC.  [VII.  78 

circular  cone  is  a  volume  that  may  be  generated  by  tbe  revolution 
of  a  right-angled  triangle  about  one  of  its  legs  as  an  axis.) 

78  Ansum-s:  3  128  cu.  ft.  4  93537284  cu.  ft.  5  386.34 
sq.  in.  6  Of  in.  7  15.919  cu.  ft.  8  1.005' J  T.  9  141.372 
sq.  ft. 

Paper  will  liave  to  be  pasted  over  the  joined  edges  of  these 
models,  that  tliey  may  liold  tlie  sand.  Let  the  pupil  measure  or 
weigh  the  sand  as  accurately  as  possible.  The  rules  which  they 
should  discover  are  as  follows  :  The  volume  of  any  pyramid  is 
equal  to  one  third  of  the  volume  of  a  prism  having  the  same  base 
and  altitude.  The  volume  of  a  cone  is  equal  to  one  third  the 
volume  of  a  cylinder  having  the  same  base  and  altitude.  From 
these  rules  may  be  derived  the  following :  The  volume  of  a  pyra- 
mid or  of  a  cone  is  equal  to  ^  of  the  area  of  the  base  X  the 
altitude.  Tlie  convex  surface  of  a  pyramid  is  the  sum  of  the 
lateral  faces.  The  convex  surface  of  a  cone  is  its  curved  surface. 
Lead  the  pupils  to  give  the  rule  for  finding  tlie  area  of  the  convex 
surface  of  pyramids  and  cones.  The  rule  is  :  Multiply  the  per- 
imeter of  the  base  of  the  pyramid  or  cone  by  one  half  the  slant 
height. 

79  Answers:  4  «  414.6912  sq. ft.;  Z*  779.1168  sq.  ft.;  c  1381.32225 
sq.  ft.  5  a  175.92  cu.  in.;  h  2865.13  cu.  ft.  8  1256.64  sq.  in. 
2827.44  sq.  in.  5  sq.  ft.  13.3+  sq.  in.  534.83+  sq.  in.  9  135.0899 
cu.  in.  4.1888  cu.  ft.  904.7808  cu.  in.  10  $1884.96.  11  512 
bullets     110609  bullets. 

The  frustum  of  a  ^^i/ra^nid  is  the  portion  of  a  pyramid  included 
between  the  base  and  a  plane  made  by  cutting  tlie  pyramid  through 
the  lateral  faces,  parallel  to  the  base. 

The  slant  heiglit  of  the  frustum  of  a  pyramid  or  of  a  cone  is 
the  perpendicular  distance  between  the  sides  of  the  upper  and 
lower  bases.  The  total  surface  of  a  frustum  is  found  by  adding 
the  surface  of  the  bases  to  the  lateral  surface.  The  lateral  surface 
is  found  by  multiplying  half  the  sum  of  the  perimeters  of  the  bases 
by  the  slant  height. 


VII.   79]  teachers'    ilANUAL.  173 

The  volume  of  the  frustum  of  a  pyramid  or  cone  is  equal  to  the 
difference  between  the  original  pyramid  or  cone  and  the  pyramid 
or  cone  cut  off.  If  the  height  of  the  original  cone  is  not  given, 
the  volume  of  the  frustum  is  found  by  the  following  rule  :  To  the 
sum  of  the  squares  of  the  diameters  of  the  two  bases  add  the  prod- 
uct of  the  two  diameters.  Multiply  this  sum  by  the  product  of 
i  of  the  height  and  .7854. 

With  an  orange  or  wooden  ball  teach  the  following  definitions  : 

A  sphere  is  a  solid  bounded  by  a  curved  surface,  every  point  oi' 
which  is  equally  distant  from  a  point  within  called  the  centre. 

The  diameter  of  a  sphere  is  a  straight  line  passing  through  the 
centre  and  having  its  extremities  in  the  surface. 

A  great  circle  of  a  sphere  is  a  section  made  by  cutting  the  sphere 
through  the  centre. 

A  good  method  of  finding  the  surface  of  a  sphere  is  to  compare 
it  with  the  surface  of  a  right  cylinder  whose  heiglit  and  diameter 
of  base  are  exactly  equal  to  the  diameter  of  sphere.  By  wrapping 
the  sphere  and  cylinder  with  narrow  waxed  tape,  and,  after  un- 
wrapping them,  comparing  the  amounts  of  ta})e,  it  will  be  observed 
that  the  areas  of  the  surface  of  the  sphere  and  the  lateral  surface 
of  the  cylinder  are  alike.  Since  the  lateral  surface  of  a  cylinder  is 
found  by  multiplying  the  circumference  of  the  base  by  the  altitude, 
it  will  be  seen  that  the  surface  of  a  sphere  is  found  by  multiplying 
the  circumference  by  the  diameter. 

To  find  the  volume  of  a  sphere,  compare  the  contents  of  the 
sphere  with  the  contents  of  the  enveloping  cylinder.  This  may 
be  done  by  making  a  ball  which  will  exactly  fit  a  cylinder.  Fill 
the  cylinder  with  water  and  immerse  the  ball.  Compare  the  depth 
of  water  before  the  ball  was  immersed  with  the  depth  of  water 
after  the  ball  is  taken  out,  and  it  will  be  found  that  tAvo  thirds  of 
the  water  has  been  displaced.  Therefore,  the  volume  of  a  sphere 
is  equal  to  two  thirds  of  the  volume  of  a  cylinder  whose  diameter 
of  base  and  height  are  equal  to  the  diameter  of  sphere.  From 
this  fact  may  be  derived  the  rule  :  Multiply  the  cube  of  the  diameter 
by  .5236  j  or,  by  conceiving  a  series  of  pyramids  formed  from  the 


174  GRADED   ARITHMETIC.  [VII.  80 

sphere,  having  their  vertex  at  the  center,  it  will  be  seen  that  the 
volume  of  the  pyramids  composing  the  sphere  must  equal  the  surface 
of  their  bases  multiplied  by  -^  of  their  radius  ;  hence  the  rule  : 
To  find  the  volume  of  a  sphere,  multiply  the  surface  by  ^  of  the 
radius. 

80  Answers:  3  90°  60°  30°.  5  69j  mi.  6  12430.8  mi. 
7  598.1  mi.  2392.4  mi.  10765.8  mi.  21531.6  mi.  8  About 
69^  mi.       9  47°     3250|  mi.       10  43°     2974i  mi. 

If  necessary,  review  previous  work  in  circles  and  degrees,  pages 
14,  15,  Book  VI. 

81  Ansjvers:  2  240'  14400".  3  640'.  4  15600".  5  495' 
405'     10'    950'.       6  4200"    15630"    22320"    900"    10".       7  600" 


12"     7650 

'.      8 

14800" 

192"    2340' 

"     21" 

9     J3  7    ° 
^360 

1 

80 

180 

10  lOf 

.061°. 

11  .75/g°     .01i°. 

12 

10°  50'  40". 

13  36° 

31'  30". 

14  9° 

34'  44 

19  23° 

10'. 

20  12°  20'. 

21  89° 

55'  22". 

S2    Ansivers : 

1  6°. 

2  llf  da. 

3  73|a  h.      4  2° 

31' 

'.      5  7' 

59f||". 

9  360^ 

'     15° 

15'.       10  3° 

45°. 

13  40  min. 

26  min. 

I  h.  41  min.  20  sec.     14  Slower    4  min.      15  11.03  p.m.    1.57  p.m. 
16  132°  10'  E. 

83  Ansivers:  2  N.Y.,  7  h.  3  min.  44  sec.  a.m.  Chi.,  6  h.  9  min. 
28+  sec.  A.M.  KO.,  5  h.  57  min.  23+  sec.  a.m.  London,  11  h. 
59  min.  37  sec.  a.m.  Paris,  12  h.  9  min.  33+  sec.  p.m.  Boston, 
7  h.  15  min.  46  sec.  a.m.  Wash.,  6  h.  51  min.  57  sec.  a.m.  Eome, 
12  h.  49  min.  4+  sec.  p.m.  Berlin,  12  h.  53  min.  34+  sec.  p.m. 
San  F.,  3  h.  50  min.  26+  sec.  a.m.  Cal.,  5  h.  53  min.  20  sec.  p.m. 
St.  L.,  5  h.  59  min.  a.m.  3  Rome,  9  h.  57  min.  50+  sec.  p.m. 
Paris,  9  h.  17  min.  22+  sec.  p.m.  Cal.,  3  h.  1  min.  22  sec.  a.m. 
5  15°.  6  21°  15'.  7  5°  East.  8  9  h.  3  min.  8/^  sec.  9  5  h. 
50  min.  8|  sec.       10  20°.     11  93°  20'  West.       12  13°  22'  6". 

84  Ansivers:    2  Ind.,  12  o'clock.    Pitts.,  1  o'clock  p.m.    Denver, 

II  o'clock  A.M.    San.  E.,  10  o'clock  a.m.    N.  0.,  12  o'clock.    Boston, 
1  o'clock  p.m.     Omaha,  12  o'clock.       3  San  Jose,  1  o'clock  p.m. 


VII.  85]  TEACaERS'    MANUAL.  175 

Portland,  4  o'clock  p.m.  Springfield,  3  o'clock  P.M.  4  Charleston, 
10  h.  15  min.  a.m.  Sac,  7  h.  15  min.  a.m.  5  Cin.,  22  min.  19  sec. 
Wil.,  11  min.  30  sec.  St.  P.,  12  min.  20  sec.  7  1  min.  16  sec. 
past  4     28  min.  20  sec.  past  1. 

85  The  relation  or  ratio  of  6  blocks  to  2  blocks  is  found  by- 
dividing  6  blocks  by  2  blocks  =  3.  The  ratio  is  expressed  thus: 
6  blocks  :  2  blocks.  The  antecedent  of  this  ratio  is  6  blocks,  and 
the  consequent  is  2  blocks. 

Lead  the  pupils  to  make  the  following  statements  from  examples 
called  for  in  11 : 

If  both  terms  of  a  ratio  are  multiplied  by  the  same  number,  the 
terms  are  larger,  but  the  ratio  remains  unchanged. 

If  both  terms  of  a  ratio  are  divided  by  the  same  number,  the 
terms  are  smaller,  but  the  ratio  remains  unchanged. 

86  A  proportion  is  the  equality  of  ratios.  The  first  and  fourth 
terms  of  a  proportion  are  called  the  extremes.  The  second  and 
third  terms  are  called  the  means.  From  observation  the  pupils 
will  discover  and  state  the  conclusion  that  the  product  of  the 
fourth  term  by  the  number  of  units  in  the  first  term  equals  the 
product  of  the  third  term  by  the  number  of  units  in  the  second 
term  ;  or,  more  briefly,  the  product  of  the  extremes  equals  the 
product  of  the  means.  Prom  tins  fact  it  will  appear  that  the  prod- 
uct of  the  extremes  divided  by  a  mean  equals  the  other  mean  ; 
and  that  the  product  of  the  means  divided  by  an  extreme  equals 
the  other  extreme. 

87  Answers:  4  «  $8  ;  h  $90;  c  f  .25  ;  d  $.52|;  e  12  lb.; 
/  16|  lb.;  rj  $625;  h  $2.13^;  i  $.47^;  ./  $18.75.  5  a  $.10-}^; 
b  $.13^;  c  $li;  d  $1.50;  e  $.15;  /$200;  r/  31b.;  h  6  ex.;  i  $.07|; 
j  $.60.       6  12/.       7  $4.80.       8  $.80.       9  3  doz.       10  $.50. 

The  last  five  problems  on  this  page  and  all  the  problems  on  the 
five  following  pages  are  intended  to  be  performed  by  analysis  and 
by  proportion.  The  solution  by  analysis  should  be  either  oral 
or  on  a  line.  The  solution  by  proportion  may  be  as  follows  (6) : 
The  ratio  of  2  apples  to  6  apples  must  be  the  same  as  the  cost  of 


176  GRADED   ARITHMETIC.  [VII.  88 

2  apples  to  the  cost  of  6  apples.  It  is  required  to  find  the  cost 
of  6  apples  ;  therefore  we  make  4  cents,  the  cost  of  2  apples,  the 
3d  term.  Since  the  greater  the  quantity  the  greater  the  cost,  we 
make  6  the  2d  term  and  2  the  1st  term.  The  proportion  is 
2:6  =  4/  :  x.  Multiplying  the  4  by  6  and  dividing  the  product 
by  5,  we  have  for  the  fourth  term  12.     Answer^  12/. 

88  Answers:  1  20  apples.  2  $80.  3  25  bu.  4  $14.25. 
5  66|  mi.  6  $22.40.  7  $1120.  8  145^  A.  9  1488  bu. 
10  450  rd.       11  2400  T.      12  5.']^  da.       13  4i  da.      14  $13230. 

15  20  ft.     16  lllj.     17  480  men.     18  29iV  mi-     19  l^-^A  ^^■ 

The  solution  on  a  line  may  be  as  follows  (13) : 
2 

%  da.  X  12      _       ^^  , 

^TT =  V  ="  H  da. 

5 

If  12  men  mow  the  meadow  in  8  days,  it  will  take  1  man  12  times 
as  many  days  to  mow  it  as  it  takes  12  men,  and  20  men  will  mow 
it  in  one  twentieth  as  many  days  as  it  takes  1  man.  Multiplying 
and  dividing,  we  have  an  answer  of  4|  days. 

89  Answers :  1  461-/3  mi.  2  $2346|.  3  13|  yd.  4  12|/. 
5  $1105.92.  6  GGOOO  times.  7  277  da.  18  h.  40  min.  8  258000 
men.  9  lOia  rolls.  10  18f  Ih.  11  1152  mi.  16f  h. 
12  $677.00+.       13  $32000.  14  444^  lb.       15  1  lb.  14+  oz. 

90  Answers:  1  4/  and  8/.  2  30  yr.  and  10  yr.  3  ftr- 
4  A,  $8;  B,  $16.  5  A,  $12  ;  B,  $10  ;  C,  $8.  6  11/,  22/,  33/. 
7  $186|,  $280,  $373^.  8  A,  $1008  ;  B,  $504  ;  C,  $168.  9  12, 
8,  4.  10  315  and  525.  11  18  rd.  and  24  rd.  12  Each  daughter, 
18000  ;  each  son,  $6000.     13  18  rolls.     14  3389|a  lb.      15  6|  A. 

16  4352  lb.       17  125  lb. 

In  sucli  problems  as  6,  the  whole  number  will  be  the  sum  of 
the  parts  1,  2,  3,  and  may  be  represented  by  6.  The  proportions 
will  be,  6  :  1  =  66  :  11  ;  6  :  2  =  66  :  22  ;  6  :  3  =  66  :  33.  By 
analysis  the  solution  is,  66  =  f ;  |  =  11 ;  |  =  22 ;  |  =  33. 


VII.  91]  teachers'  manual.  177 

In  9,  reduce  the  fractions  to  a  common  denominator  and  nse  the 
nnmeratorSj  as  in  6. 

91  Ansivers :  1  A,  ^^00  B,  .flSOO.  2  A,  14800  P>,- f  2880 
C,  $4320.  3  A,  $G35yV  T.,  $204jf.  4  A,  $1800  B,  $2700. 
5  Hall,  $888  Eeed,  $5i)2.  6  A,  $3777?,  B,  $5(3G()|  G,  $7555^ • 
7  A,  $G26,2^  B,  $1173M.  8  A,  $()4,^,  B,  $43J.j  C,  $32^3. 
9  A,  $49t^V  B,  $30|!i.  10  A,  $15600  B,  $20400  C,  $24000. 
11  A,  $3840     B,  $4800     C,  3300. 

In  7  to  11,  make  the  conditions  uniform  before  the  proportion 
is  made.  A's  share  of  the  work  done  was  equal  to  the  service  of 
192  men  1  day,  and  B's  share  was  360  men  1  day.  A's  share  of 
the  contract  money  should  be,  therefore,  i^f  of  $1800. 

92  Answers:  1  5^  da.  2  2  gal.  3  9^  weeks.  4  320  mi. 
21?  da.  5  5  mo.  6  $240.  7  Hi  da.  8  370i?  rd.  9  272^12 
I765L.  10  T)!  rd.  11  8/  2/  6  of  40,^  and  4  of  50/.  12  3  of 
6/  and  1  of  8/     3  of  6/  and  5  of  8/. 

These    problems   should   be    performed  by  analysis   on    a  line, 

rather  than  by  compound  proportion.     The  solution  of  8  may  be 

as  follows  : 

40  5 

$(l)  rd.  X  M  X  50       -„  ,      , 
n :r: —  =  3  i  ^K  rd. 

X$   X   M 

9  3 

My  answer  is  to  be  in  rods,  therefore  80  rods  is  the  number  to 
work  upon.  1  ]nan  will  build  ^l  as  many  rods  as  18  men,  found  by 
dividing  by  18  ;  40  men  will  build  40  times  as  many  rods  as  1 
man,  found  by  multiplying  by  40.  If  they  can  build  so  many  rods 
in  24  days,  in  1  day  they  will  build  ^\  as  many  rods,  found  by 
dividing  by  24,  and  in  50  days  they  will  build  50  times  as  many 
rods  as  they  build  in  1  day,  found  by  multiplying  by  50.  Simpli- 
fying by  cancellation  wo  have  372V  ^'^■ 

93  Nearly  all  of  these  problems  should  be  performed  orally. 

94  A7isive)'s:  7  1988515.54  T.  8  311694000001b.  15584700 T. 
9  $31684702.80.       10  W.,  46.66%     CI.,  56.89%     N.Y.,  44.06% 


178  GKADED   ARITHISIETIC.  [VII.  95 

Bos.,  72.13%  L.,  39.34^0  Ph.,  62.71%  Bal.,  25%  S.L.,  41.37% 
Br.,  62.71%     Cin.,  55.17%     Pitts.,  46.77%     Cli.,  47.36%. 

95  .  A7iswers :  1  257^7  ft.  2  ^hVo-  3  ^6  IQi  30.  4  66f% 
33^%  400%.  5  5  3  2000  .002.  6  .08  .0003  .00001 
2400.       7  200  bolts     4000  bolts     21100  bolts     $3     $3.46     $.58. 

8  $24.50  $58.19  $19.69.  9  $24.75.  10  $50.63.  11  $1513.51. 
12  70%.  13  $10200.  14  95  shares  ($235  left).  15  34.6+%. 
16  52  shares  ($80  left).       17  $88500.       18  $300.       19  8f  %. 

96  Answers:  1  116  bales.  2  $3037.50.  3  $7507.58. 
4  $3461538.46.     5  $25.35.     6  $69.23.     7  42  shares.     8  28+%. 

9  $7087.50.  10  $10675  $97.50  and  $154.32.  11  $10008. 
12  $40099.75. 

97  Ansivers:  1  $1500.  2  $362.09.  3  $7200000.  4  3  h. 
44  m.  P.M.  5  $301.50.  6  $807.50.  7  3^%.  8  $4473.79 
$192     $134.21.       9  $111.23.        10  1  yr.  21  da.        11  $987.65. 

12  $100. 

98  Ansivers:     1    $393.89.         2     $685.94.         3    $4125.72+. 

4  $4800.  5  80%.  6  $925.  7  .00948.  8  $1550  $1475 
$1400     $1325.       9  $6744.36.       10  $112.31+     $75     5.3+%. 

99  Answers:    1    $8.22.         2    $26.44.         3    $110.10.        4   5. 

5  $496.13.  6  $60.  7  $106.55/j.  8  $385.71f  9  960 
stones.        10  2-j\  da.        11  6oi  yd.     $122.50.       12  380  sq.  mi. 

13  4945|  sec.     A  little  more  than  ^  of  a  second.       14  $87.22. 

100  A7iswers:  1  $432.56.  2  5616  tiles  2990  tiles.  3  $3.62^ 
4  $1441.19.  5  $1438.25.  6  76/.  7  54^6_  ^^k.  8  $18.13. 
9  (a)  $2450.25;  (b)  20f|f  cu.  yd.;  (c)  $167.24+;  (d)  $36.16; 
(e)  $58.58  ;  (/)  2.019+  bu.       10  2538.0864  gal.     $45.24. 

101  Ansivers:  1  6f|  cu.  yd.  2  20  yd.  3  72000  cu.  ft. 
120  ft.  4  25  42.24 '^"\  5  2.891  +  '"  .18™'".  6  1.8°  P. 
|°C.  5fC.  25°  C.  68°  P.  95°  P.  7  37^°  C.  8  20°  C.  9  38400 
.cu.  ft.     512000  cu.  ft.    .  10  .655+  gal.       11  8.6+  qt. 

103  Answers:  1  548  ft.  2  12.04+  sec.  3  1.27+  sec. 
4  23209140800000000  miles.  5  10  lb.  6  20  lb.  7  3400  lb. 
7000  lb.  8  2|  and  1.  9  5^  lb.  10  83^  lb.  11  If  ft.  from 
the  end. 


VIII.  1]  teachers'  manual.  179 


SECTION   X. 

NOTES   FOR   BOOK  NUMBER   EIGHT. 

Unless  the  principles  and  processes  of  the  various  subjects  of 
Arithmetic  are  familial*,  it  will  be  necessary  to  give  some  prelim- 
inary reviews  before  the  pupils  are  able  to  give  the  definitions  and 
rules  called  for  in  this  book.  The  exercises  themselves  suggest 
some  desirable  kinds  of  review  work,  as  well  as  some  forms  of 
definitions  which  may  be  made.  For  suggestive  hints  as  to  methods 
of  teaching  definitions  and  rules  see  Manual,  pages  139,  140. 

Some  portions  of  the  section  on  Geometry  may  seem  too  difficult 
for  pupils  of  Grammar  grades,  but  if  the  geometrical  exercises 
of  previous  books  have  been  fairly  well  performed,  pupils  of  the 
highest  grades  should  be  able  to  take  all  the  inventional  and  con- 
structive work,  much  of  which  is  a  review,  and  the  easier  portions 
of  the  demonstrative  work.  The  most  difficult  exercises  might 
be  given  to  such  pupils  only  as  show  a  special  aptitude  for  the 
study  of  geometry.  In  this,  as  in  other  subjects,  the  aim  should 
be  to  give  work  that  shall  be  sufficient,  both  in  amount  and  kind, 
to  fully  tax  the  powers  of  every  pupil. 

1  In  4,  lead  the  pupils  to  discover  the  fact  that  the  decimal 
system  of  numbering  depends  upon  the  principle  that  ten  units  of 
one  order  equal  one  unit  of  the  next  higher  order.  Other  systems 
should  be  illustrated  by  examples.  In  the  duodecimal  system,  for 
example,  the  pupils  should  be  led  to  see  that  ^,  -J,  ^,  and  I,  can 
be  expressed  by  a  single  figure ;  thus,  .6,  .4,  .3,  .2  ;  and  that  ^  and 
I  can  be  expressed  by  two  figures ;  thus,  .18  and  .16. 

The  origin  of  the  words  expressing  numbers  to  twelve  may  be 
found  in  an  unabridged  dictionary.  Other  names  are  derivatives, 
and  their  origin  will  be  plainly  seen. 

The  definitions  called  for  on  this  page  are  as  follows : 

A  unit  is  anything  which  is  considered  as  one. 


180  GRADED   ARITHMETIC.  [VIII.  1 

An  integral  unit  is  one  or  a  collection  of  ones  regarded  as  an 
undivided  whole. 

A  fractional  unit  is  a  part  of  one  regarded  as  an  undivided  whole. 

A  number  is  a  unit  or  a  collection  of  units.  An  integral  numher 
is  an  integral  unit  or  a  collection  of  integral  units.  K  fractional 
numher  is  a  fractional  unit  or  a  collection  of  fractional  units. 
Number  is  a  quality  of  objects  which  answers  the  question,  ''how 
many,"  and  arises  from  distinguishing  one  from  more  than  one. 

Arithmetic  is  that  knowledge  which  has  for  its  object  the  ex- 
pression, the  operations,  and  the  relations  of  numbers. 

The  sum  of  two  or  more  numbers  is  their  united  value. 

Addition  is  the  process  of  finding  the  sum  of  two  or  more 
numbers. 

Subtraction  is  the  process  of  taking  away  a  part  of  a  number  to 
find  how  many  units  are  left.  The  minuend  is  the  number 
separated.  The  subtrahend  is  the  number  taken  away.  The 
remainder  is  the  part  left. 

Multij)lication  is  the  process  of  finding  the  united  value  of  two 
or  more  equal  numbers.  The  multi^jlicand  is  the  number  multi- 
plied. The  mtdtiplier  is  the  number  which  shows  how  inany  times 
tlie  multiplicand  is  taken.  The  pi'oduct  is  the  result  obtained  by 
multiplication.  The  factors  of  a  number  are  the  numbers  which, 
multiplied  together,  produce  that  number. 

Division  is  the  process  of  finding  one  of  the  equal  parts  of  a 
number,  or  of  finding  how  many  times  one  number  is  contained  in 
another.  The  dividend  is  the  number  divided.  The  dinisor  is  the 
number  by  which  we  divide.  The  quotient  is  the  result  obtained 
by  division. 

An  odd  number  is  a  number  which  cannot  be  separated  into  two 
equal  integral  parts.  An  even  numher  is  a  number  which  can  be 
separated  into  two  equal  integral  parts. 

A  prime  mimher  is  a  number  whose  only  integral  factors  are 
itself  or  one.  A  composite  number  is  a  number  which  is  composed 
of  other  integral  factors  besides  itself  or  one. 

K  prime  factor  is  a  factor  which  is  a  prime  number. 


VIII.  2]  teachers'  manual.  181 

A  multiple  of  a  number  is  any  whole  number  of  times  a  number. 
A  common  multiple  of  two  or  more  numbers  is  a  number  wliich  is 
a  multiple  of  each  of  the  numbers.  Tlie  least  common  multiple  of 
two  or  more  numbers  is  the  least  number  wliieh  is  a  multiple  of 
the  numbers. 

A  common  divisor  or  common  factor  of  tAVO  or  more  numbers  is 
a  factor  which  belongs  to  each  of  them.  The  greatest  common 
divisor  or  factor  of  two  or  more  numbers  is  the  greatest  factor 
which  belongs  to  each  of  them. 

The  denoyninator  of  a  fraction  is  the  number  of  equal  parts  into 
which  the  unit  is  divided.  The  numerator  is  the  number  of  equal 
parts  taken. 

To  change  fractions  to  equivalent  fractions  having  the  least 
common  denominator  :  Divide  the  least  common  multiple  of  the 
denominators  by  the  denominator  of  each  fraction  and  multiply 
both  terms  of  the  fraction  by  the  quotient. 

To  add  fractions  :  Change  the  fractions  to  equivalent  fractions 
having  a  common  denominator  ;  add  the  numerators,  and  write  the 
sum  over  the  common  denominator. 

3  To  subtract  fractions :  Change  the  fractions  to  equivalent 
fractions  having  a  common  denominator,  and  write  the  difference 
of  the  numerators  over  the  common  denominator. 

To  multiply  a  fraction  by  an  integer  :  Multiply  the  numerator 
of  the  fraction  by  the  integer  and  place  the  product  over  the 
denominator  ;  or,  divide  the  denominator  of  the  fraction  by  the 
integer  and  place  the  quotient  under  the  numerator. 

To  find  the  fractional  part  of  a  fraction  :  Write  the  product  of 
the  numerators  over  the  product  of  the  denominators. 

To  divide  an  integer  or  fraction  by  a  fraction  :  Change  the 
numbers  to  equivalent  fractions  having  a  common  denominator, 
and  divide  as  in  Avhole  numbers  ;  or,  invert  the  divisor  and  proceed 
as  in  multiplication. 

The  surveyor's  chain,  often  called  Gunter's  chain,  has  largely 
gone  out  of  use,  having  been  replaced  by  the  steel  ribbon,  com- 
monly 100  feet  long,  with  the  principal  divisions  and  markings 


182  GRADED    ARITHMETIC.  [VIII.  3 

at  the  foot  points,  with  other  divisions  at  tenths  and  hundredths 
of  feet. 

Other  points  called  for  may  be  found  in  the  Tables  given  at  the 
end  of  Book  VII.  or  in  the  Appendix  of  Book  VIII. 

3  The  standard  unit  of  dry  measure  is  the  bushel,  which  con- 
tains 2150.42  cu.  in.  This  bushel  is  sometimes  called  the  Win- 
chester bushel,  which  was  the  standard  bushel  of  England  before 
the  imperial  bushel  was  adopted.  The  imperial  bushel  contains 
2218.192  cu.  in.,  or  80  lb.  of  distilled  water.  The  standard  unit 
of  liquid  measure  is  the  gallon,  which  contains  231  cu.  in.  A 
chaldron,  sometimes  used  in  measuring  charcoal,  contains  36  bu. 
Barrels,  hogsheads,  tierces,  and  pipes  vary  in  capacity.  A  stone, 
sometimes  used  in  measuring  iron  and  lead,  is  14  lb. 

The  meter  is  the  standard  unit  from  which  all  metric  measures 
are  derived.  It  is  intended  to  be  the  ten-millionth  part  of  the 
distance  on  a  meridian  from  the  equator  to  the  pole.  Lead  the 
pupils  to  see  the  connection  of  all  parts  of  the  metric  system  — 
a  cubic  centimeter  of  water  (at  39°  F.)  weighing  1  gram,  and  a 
cubic  decimeter  of  water  measuring  1  liter. 

A  solar  year  is  the  average  time  it  takes  the  earth  to  make  a 
complete  revolution  around  the  sun,  or  about  365  da.  5  h.  48  luin. 
49  sec.  This,  it  will  be  seen,  is  about  365:^  days,  and  therefore 
one  extra  day  is  counted  every  fourth  year.  But  by  this  reckoning 
we  should  gain  nearly  a  day  in  one  hundred  years,  and  therefore 
the  centennial  years  are  commonly  counted  as  ordinary  years.  If 
all  the  centennial  years  were  thus  counted,  we  should  lose  nearly 
a  day  in  four  hundred  years  ;  therefore,  only  the  centennial  years 
divisible  by  400  are  counted  as  leap  years.  It  would  be  well  in 
this  connection  to  tell  the  pupils  how  and  when  this  new  style  of 
reckoning  the  years  was  adopted. 

The  lunar  month,  is  the  time  between  two  new  moons,  and  is 
about  29  da.  12  h.  44  min.  long. 

All  other  measures  and  values  referred  to  on  this  page  may  be 
found  in  the  Appendix. 

4  The  percentage  is  a  number  which  is  a  certain  number  of 
hundredths  of  another  number. 


VIII.  5]  teachers'  manual.  183 

The  hoie  is  the  number  of  whicli  a  number  of  hundredths  is 
taken  tu  hnd  the  percentage. 

The  rate  is  the  number  of  liuudredths  which  the  percentage  is  of 
the  base. 

The  amount  is  the  sum  of  the  base  and  percentage. 

Tlie  remainder  is  the  difference  between  the  base  and  percentage. 

Lead  the  pupils  to  find  these  terms  in  Profit  and  Loss,  Insur- 
ance, Duties,  Interest,  etc.,  and  to  give  examples  as  required  in 
2  to  4. 

Other  points  referred  to  on  this  page  are  explained  in  Section 

IX.  of  the  Manual. 

5  In  2,  the  formula  is  B^P-^B.  In  3,  B  =  A  ^  (1  +  ^). 
"While  it  is  not  advisable  generally  for  pupils  to  perform  problems 
in  percentage  by  formulas,  it  is  good  practice  occasionally  to  per- 
form them  in  that  way,  especially  when  the  formulas  are  made  by 
the  pupils. 

6  Numbers  may  be  added  in  pairs  instead  of  one  at  a  time  ; 
thus,  in  adding  28,  36,  43,  84,  68,  74,  either  in  a  column  or  in  a 
line,  the  process  might  be  12,  19,  33  ;  10,  24,  31,  33.  Ansiver,  333. 
If  there  are  many  numbers  to  be  added  they  might  be  divided  into 
sections  and  the  separate  sums  added.  Two  columns  may  be 
added  at  once  by  first  adding  the  column  of  the  higher 
denomination  ;  for  example,  in  adding  the  numbers  of  this  48 
column  the  addition  might  be  made  as  follows  :  45,  105,  37 
109,  129,  134,  164,  171,  211,  219.  25 

In  6,  let  the  pupils  see  that  multiplication  by  the  aliquot         64 
parts  of  10,  100,  and  1000  may  be  made  by  annexing  one,         45 
two,  or  three  ciphers  and  dividing.      To  multiply  by  any 
number    between  91  and  99,  multiply  by  100  and  subtract.    The 
same  process  may  be  followed  in  multiplying  by  99,  999,  etc. 

In  dividing  by  the  aliquot  parts  of  100,  first  divide  by  100,  and 
then  multiply. 

To  add  fractions  having  1  fur  the  numerator,  place  the  sum  of 
the  denominators  over  their  })roduct.  To  multiply  any  number 
containing  ^,  by  itself.     Multiply  the  Avhole  numbers,  and  to  the 


184 


GRADED    ARITHIVIETIC. 


[VIII.  7 


product  add  ^  of  the  sum  of  the  whole  number  and  ^ ;  for 
example  :  16^  X  16^  =  16  X  16  =  256  +  (^  of  32)  +  ^  =  272^. 
From  this  process  lead  the  pupils  to  make  a  general  rule  for  the 
multiplication  of  mixed  numbers  in  which  the  fractions  are  alike. 
To  multiply  mixed  numbers  when  the  whole  numbers  are  alike: 
Multiply  the  whole  numbers,  and  to  the  product  add  that  part 
of  one  of  them  expressed  by  the  sum  of  the  fractions,  and  to  this 
sum  add  the  product  of  the  fractions  ;  thus  :  16f  X  16:^^  16  X  16 
=  256  ;  f  of  16  =  10  ;   f  X  i  =  ^\.     256  +  10  +  ^^^  =  266  j\,  A?is. 

7  These  exercises  are  intended  to  be  performed  with  the  fewest 
possible  figures  by  the  shortest  method.  Let  the  pupils  practice 
upon  them  until  facility  is  acquired. 

8  Atisivers :  1  2  yr.  2  mo.  12  da.  2  $62.50.  3  $90  more. 
4  6487.15+  bu.  5  7^^^%.  6  $300.  7  $4565.34  e^W/^W, 
or  about  6^%.       8  Atch.  the  better  by  ^ff  %. 

9  Answers:   1  $8666f.       2  30%.       3  200%    20  men    7  men. 
4  9.45  cu.  in.       5  59f|  qt.     5la  qt.       6  538f f  gal. 
8  2m  ft.     9  8i  ft.     10  820i  T.     11  4712^2_9_  p^t. 
75  lb.     80  lb. 

10  Answers 


7  lOf  f  in. 
12  66|lb. 


44|  lb. 


Languages. 

Number  of  Persons  Spoken  by. 

Percentage 

of 

Increase  in 

Eighty-nine 

Years. 

Percentage  of  the 
Whole. 

1801. 

1890. 

1801. 

1890. 

English         

20520000 
31450000 
30320000 
15070000 
261(»0()00 
7480000 
30770000 
161800000 

111100000 
51200000 
75200000 
33400000 
42800000 
13000000 
75000000 

441.4 

62.7 
115.03 
180.91 
157.0 
171.7 

71.8 

12.7 
19.4 
18.7 

9.3 
16.2 

4.7 

in.o 

27.7 
12.7 
18.7 

8.3 
10.7 

3.2 
18.7 

French  

German 

Italian 

Spanish 

Portuguese!  

Russian 

Total 

401700000 

248.26 

100.0 

100.0 

2  $2750.      3  36.     4  June  21,  5  h.  23  min.  30  sec  p.m.     5  $24.53. 
6  About  12  acres. 


VIII.  11]  TEACHEKS'    MANUAL.  185 

11  Ans7vers:  1  0%.  2  60%.  3  ^25500.  4  $7522.50. 
5  «i25000.  6  3/.  7  $12000.  8  if!  12315  i8;35685.  9  $243.20. 
10  $9500.624-     $5179.74.     11  C.  &  S.  M.,  2f  %.     12  $19600. 

12  Ansivers:  1  $2900.87.  2  $506.54.  3  2666|  lb.  sugar 
cane  ;  4000  lb.  beet  root ;  5000  lb.  wheaten  flour.  5  0  100°  C. 
33^°  C.  44f  C.  68°  F.  6  61.952  gr.  7  732.  8  Uf  C. 
9  Germany,  27.6+%  England,  12.3+ %  Ireland,  12.5+%  Nor- 
way and  Sweden,  10'.6+%.       10  2.2+%     42.2+%     17.8+%. 

13  The  pupils  should  be  led  to  see  by  exercises  similar  to  those 
given  on  this  page  that  symbols  denoting  quantity  may  be  expressed 
by  figures  and  by  letters,  and  that  operations  may  be  denoted  by 
signs.  The  knowledge  that  pupils  possess  of  the  expression  of 
numbers  by  figures  and  their  operations  by  signs  should  be  used  in 
leading  them  to  acquire  a  knowledge  of  the  expression  of  algebraic 
quantities  and  operations.  Such  knowledge  will  help  them  to  make 
generalizations  in  number  and  to  solve  problems  that  cannot  be 
solved  easily  by  the  aid  of  figures. 

14  Pupils  will  probably  be  able  to  perform  the  first  ten  exer- 
cises with  little  difficulty.  If  any  difficulty  is  found,  let  figures 
be  substituted  for  letters  in  representing  number,  and  let  the  num- 
ber of  exercises  be  increased. 

In  clearing  equations  of  fractions,  lead  the  pupils  to  see  the 
principle  involved  by  using  figures  instead  of  letters.  Tlie  fact 
that  multiplying  both  sides  of  the  equation  by  the  same  number 
does  not  affect  the  equality  may  be  shown  by  multiplying  equal 
numbers  by  the  same  number  and  letting  the  pui)ils  see  that  the 
products  are  equal. 

In  9,  the  axiom  that  equal  quantities  divided  by  the  same  or 

equal  quantities  give  equal  quotients  may  also  be  shown  by  the 

use  of  figures  ;  thus  : 

20  +  10      30 
20 +  10  ==30  — -^ —  =  ^ 

Other  axioms  may  be  shown  in  a  similar  way  as  they  are  needed. 


186  GRADED    AKITHMETIC.  [VIII.  15 

15  Answers:  1  Horse,  $288  Cow,  $72.  2  A,  $100  B,  $50 
C,  $150.  3  Eobert,  40  William,  20  Thomas,  120.  4  9 
oranges  and  9  bananas.  5  12.  6  Father,  48  yr.  Son,  24  yr. 
7  8  lb.  8  4  lb.  Mocha  12  lb.  Java.  9  A,  $6000  B,  $3000 
C,  $1000.       10  A  has  $G00. 

The  following  form  of  analysis  is  suggested  for  this  class  of 
problems  : 

1    Let     X  =  cost  of  cow.  • 

4 03=    "      ''  horse. 
5  .X  =    "      "       <'      and  cow. 
5  a;  =  $360. 

X  =    $72,  cost  of  cow. 
4.r  =  $288,  cost  of  horse. 

The  pupils  should  learn  to  select  the  number  that  can  be  most 
conveniently  represented  by  x.  For  example,  in  2,  the  number  of 
dollars  that  B  has,  and  in  3,  the  number  of  marbles  that  William 
has,  is  the  most  convenient  unit  of  representation. 

In  simplifying  11  to  15,  lead  the  pupils  to  see  how  the  paren- 
theses may  be  removed  in  such  examples  as  :  12  +  (4  +  2);  (8  +  5) 
+  (6  +  2);  (8 -6) +  (8 -4);  16 -(8 -6);  10  -  (6  +  2).  It 
may  be  shown  that  the  10  —  (8  —  6)  is  equal  to  16  —  8  +  6  by 
first  subtracting  8  from  16  as  indicated,  and  calling  attention  to 
the  fact  tliat  we  have  subtracted  a  number  too  large  by  6,  and 
therefore  we  must  add  6  to  the  difference.  In  the  expression 
10  —  (6  +  2)  by  subtracting  6  instead  of  6  +  2,  we  have  a  subtra- 
hend too  small  by  2,  and  therefore  we  must  subtract  2  from  the 
difference. 

16  The  four  axioms  involved  in  5  to  8  are  as  follows  : 

If  the  same  quantity  or  equal  quantities  be  added  to  equal 
quantities  the  sums  will  be  equal 

If  the  same  quantity  or  equal  quantities  be  subtracted  from  equal 
quantities  the  remainders  will  be  equal. 

If  equal  quantities  be  divided  by  the  same  quantity  or  equal 
quantity  the  (piotients  will  be  equal. 


A^lli.  17]  teachers'  manual.  187 

Two  quantities  each  ('(^iial  to  a  tliii-d  (piniitity  are  equal  to  each 
other. 

The  kind  of  illustrative  problems  called  for  is  shown  in  the 
following  : 

James  has  18  cents,  which  is  3  cents  more  than  John  has.  How 
many  cents  has  John  ? 

Let  X  =  the  number  of  cents  that  John  has. 

X  -{-S  =  18,  tlie  number  of  cents  that  James  has. 

Subtracting  3  from  these  equal  quantities  and  there  is  given : 

a;  +  3  —  3  =  18  —  3. 
.r  ^15. 

Lead  the  pupils  to  give  and  to  solve  similar  problems  illustrating 
the  four  principles. 

17  A7iswers:  16  8  6  15.  17  12  10  4.  18  12  8  9. 
19  12     18     30.       20  8     24     15. 

In  removing  the  parentheses  (11  to  15),  lead  the  pupils  to  see 
the  reason  for  changing  the  sign  by  asking  the  following  questions  : 
"  In  11,  what  is  to  be  subtracted  from  20  ?  If  6  alone  is  subtracted, 
is  the  result  larger  or  smaller  than  the  required  ansAver  ?  What 
more  must  be  subtracted  ?  How  may  both  processes  be  indicated  ?  " 
In  the  same  way  proceed  with  other  exercises  until  the  pupils  can 
see  why  parentheses  may  be  removed  by  changing  the  signs  of  all 
except  the  first  quantity. 

18  Ansn-ers:  1  Corn,  $1.42  ;  wheat,  f  1.54.  2  James,  26  yr.; 
sister,  16  yr.  3  Robert,  12/;  James,  24/.  4  34,28,20.  5  18, 
18,  24.  6  8  lb.  @  7/,  12  lb.  @  5/.  7  7,  8,  9,  10.  8  16  and 
24.  9  24.  10  48.  11  40.  12  Robert,  96/;  Ralph,  72/. 
13  180  A.       14  18  yr.       15  10,  8. 

19  Answers:  1  Father,  45  yr.;  son,  ]8yr.  2  $1200,  f!1800, 
$2500.  3  .4,  $4000;  5,  $6000  ;  C,  $!)0()0.  4  $.250.  5  13^  lb. 
6  800.  7  43t\.  8  640.  9  40.  10  $5.  11  8/.  12  $800. 
13  $4800.       14  $8000.       15  $900.       16  $70.       17  $4000. 


188  GRADED   AHITHMETIC.  [VlII.  20 

20  The  theory  of  negative  quantities  may  be  shown  by  the 
device  indicated  at  the  head  of  the  page.  In  the  relative  series 
indicated,  all  the  quantities  may  be  said  to  increase  by  1,  or  by  a 
from  left  to  right,  and  to  decrease  by  1,  or  by  a  from  right  to  left, 
the  positive  quantities  being  at  the  right  of  zero,  and  the  negative 
quantities  being  at  the  left  of  zero.  The  device  may  be  extended 
in  illustrating  1,  the  pupils  being  asked  to  begin  at  the  zero  point 
and  pass  the  pencil  to  the  right  4  spaces.  This  would  be  indicated 
by  -j"  4a  ;  then  3  more  spaces.  The  distance  from  zero  Avould  now 
be  indicated  by  -|-  7a;  and  by  moving  the  j^encil  over  2  more  spaces 
the  distance  from  zero  would  be  indicated  by  -\-9a.  2  may  be 
illustrated  in  the  same  way,  tlie  pencil  in  each  case  passing  in 
the  opposite  or  negative  direction.  The  spaces  passed  over  in  all 
would  be  indicated  by — 6  a.  In  3,  the  pencil  starts  from  zero, 
as  before ;  passes  to  the  right,  as  indicated,  4  spaces  ;  then  to  the 
left  3  spaces.  The  pencil  now  is  1  space  to  the  right  of  zero,  and 
the  distance  would  be  indicated  by  +1«,  or  +  "•  The  pencil 
moves  to  the  right  2  spaces,  and  the  distance  would  be  indicated 
by  -\-3a,  Avhich  is  the  required  answer. 

Considering  subtraction  as  the  process  of  finding  the  difference 
between  two  quantities,  the  minuend  and  subtrahend  may  be  indi- 
cated in  the  relative  series  at  the  right  or  left  of  zero,  and  the 
difference  by  the  distance  between  them.  Thus,  in  4,  the  distance 
between  +  4a  and  +  3a  is  +  a.  In  5,  the  question  is,  what  quan- 
tity added  to  —  2  a  will  equal  +  4  a.  Beginning  at  —  2  a,  the  pencil 
must  move  to  the  right,  or  in  a  positive  direction,  G  spaces  before 
it  reaches +4  a.  The  difference  is,  then,  indicated  by+Ga.  In 
6,  the  pencil  must  move  7  spaces  to  the  left  from  +  3  a  to  reach 
the  point  —  4  a.     The  difference  is  therefore  indicated  by  —  7  a. 

After  several  similar  illustrations  are  given  the  pupils  should 
be  ready  to  apply  the  principle  expressed  in  7.  In  Algebra,  the 
multiplier  indicates  the  number  of  repetitions  either  of  addition 
or  subtraction  that  is  made  of  the  multiplicand.  For  example, 
(-}-  4  a)  X  (-|-  2)  means  that  -}-  ia  is  to  be  added  2  times,  and 
(-}-  4  a)  X  ( —  2)  means'  that  +  4  a  is  to  be  subtracted  2  times.     The 


VIII.  21]  teachers'  manual.  189 

answer  in  the  first  case  is  -\-  8  a,  and  in  the  second  case  —  8  a. 
Again,  adding  — 4  a  2  times,  we  have  for  a  result  — Sa,  and  sub- 
tracting —  4«  2  times,  Ave  have  +  8 '-f^j  '•t'-  (~  ■!'')  X  (+  2)  =  —  Sa, 
and  (—  4  a)  X  (—  2)  =  +  8  a. 

After  this  explanation  is  fully  understood  l)y  the  pupils,  and 
can  be  given  by  them  in  multiplying  other  quantities,  they  may 
make  use  of  the  following  rule  : 

The  sign  of  the  product  of  two  numbers  is  plus  if  the  signs  of 
the  numbers  are  alike,  and  minus  if  the  signs  are  unlike. 

Since  Division  is  the  reverse  of  Multiplication,  and  the  quotient 
is  that  number  which,  multiplied  by  the  divisor,  will  give  the  divi- 
dend, the  way  of  finding  the  sign  of  the  quotient  will  readily 
appear.  The  rule  for  both  Multiplication  and  Division  in  brief 
will  be  : 

Like  signs  give  plus  ;  unlike  signs  give  minus. 

31  Some  teaching  may  be  necessary  for  a  few  of  these  exer- 
cises, especially  those  which  involve  the  midtiplication  and  division 
of  powers  of  tlie  same  quantity.  No  attempt  should  be  made  to 
simplify  results  by  factoring  at  this  time,  but  later,  if  it  is  thought 
desirable,  the  results  may  be  reduced  to  their  simplest  forms.  In 
some  cases  of  expressed  division,  as  in  14,  (6ab)  -^  (ftc),  the  common 
factor  a  may  be  removed,  as  in  similar  cases  of  Arithmetic. 

23  Answers:  1  ((x-\-kr  f'!/-}-^!/  — <f-T  —  %  — ('.'/  —  b>/- 
2    «.i'  -|-  ki'  -\-  ((I/  -\-  hij         (tx  -\-  hi/  —  It  (/  —  /yy.  3     ".«'  —  hx  -\-  a  ij  —  hij 

ax  —  /m —  <i  tj  -\rliij.         4  cr  -f"  2  r?  -|-  1       "'  —  1        1  —  ""•  5  6 ax 

-\-A,h.i — S^'cc  — Q>axi/  —  Altx;/  -\-?>cxy.  6    4<f^j'-f- Ga"''  —  8a 

4 a\r  +  (;  (i\v  —  8 ax        4 a''x  -f  G a^  —  8 a  —  2 ax  —  3 fl^ -f  4.  7  4.a^b 

+  2 a%''  +  a/r  4 a'%  +  2  aV-  -f  ah^  —  4  a%  —  2  ah'  —  //"'  —  4  a'-h 

—  2  «2^2  _  ,,/,!  _^  4  ,,2/,  _^  2  ah'  -f  h^  8  (r  +  2  ah  +  h'  +  ar  +  he 
a'  +  ac  —  h'  —  hr       a- -}- 2ah  +  h' +  2  ac -\- r' -\-2hc.         9  2x^  —  43;^ 

+x/       2.rV-4xy  +  uy-f2xV-4V  +  /       2x'>/-4:xy  +  x2/ 

—  2x'i/-\-Axi/'  —  y^.  10  x'-\-2xi/-\-y'  x' —  if  x^-\-x^y-\-x'f-\-'i^ 
x^-\-xSi  —  xxf — if,  11  x^y-\-x'ij-\-x^f-\-xxf       x^f-\-x'if-\-x^y 

—  xyx'f-\-x'f      xhf-^x'f.  12  a;"  — 2a;^  — 13x''  +  14ic  +  24. 


190  GEADED    ARITHMETIC.  [A'lII.  23 

13   x-^2xi/-\-if      x-  —  2xy-\-ir       .r^  +  2,>'  +  l.  14  ,r--2,r  +  l 

4.r-  +  S.ry-f'±.'/  4.r2  — 8.ry  +  4/.  15  a"  +  ?> a'- -\- 5 air ^ b" 

a'-t^a-b  —  'iab'  —  lf'      rr^  +  3«-  +  3«  +  l.  16  a-b'- +  2  abed  +  vM- 

a%''  +  2ab  +  l      ./■^  +  2.r-y+//.         17  2a-  +  2a.         18  2x''+2>f. 

1:Q  2crb  +  ^aJr  +  2lf.  20  b-\-r  +  rl         ^^±j±^         ^+^^ 

^  +  ''-^'K  21  6rt  +  12/>  +  lS         (/  +  2/;  +  3         fr  +  2r^5  +  3«. 

« 

22   1      ^^  +  ^'      «-  +  2rY/>  +  /r.       23   a  —  b      a-  —  2al>  +  lr.       24   r^  +  ^' 

1.        25  «  — />      ^^  +  A.        26  r/2  +  2.//>  +  /r      r/  +  /;.        27  2r  +  l 

2x  — 1.  28   1  — 2.r  2.r  — 1.  29   />  — r         — />  +  r'  a. 

30  r  +  (/.  31  2  +  3j'  — r>.r-.  32  a-  —  lr      a-  +  b\         33  4a; 

+  3//  +  4.       34  3.r  — 2y+l.       35  3.r  +  2^. 

23    Ajxsu-ers:    1  2(5      ./ +  A  +  r      6«.         2  25      «  +  /v~c      8  a. 
3—10.  4  rt  — 4Z>  +  c+r/.  5  .r  — 3.r-.  6  7a:-  +  4a;  +  ^-. 

7  8.r  +  5ia;l  8  5.r-  +  .r +  2y  — 15.  9  .t"  +  3/  — 2a'  +  2. 

•t  r\        ^  A  i\  90  40  900  Ofio  oro  12  2  O  (O 

10  x'    X*    y^    x-y^    x*y    x-ifrJ    x-y'z-    x- -\-  lx]i-\-  y    x-  —  2xij-\-y 
a;2  +  2  a;  + 1     x* -\- 2  xhf- +  y'     9  a-  +  12  ah  +  4  //I       n  Sum  of  their 
squares  +  twice  their  product     Sum  of  tlieir  squares  —  twice  their 
product.  12    x^       x^if       x^yh^       x'^y^z^       x^-\-ox-y-\-Sxy~-\- y^ 

x'  -  3  xV  +  3  xy'-  —  ir  8  x^  +  24  a;2  +  24  x  +  8  27  j^  -  2  7 ./-  +  9  x  - 1 
64a^  +  96a2^<-f  48«i'2  +  8/>l  13  q^^^  ^f  ^he  1st  +  3  times  the 
square  of  the  1st  into  the  2d  +  3  times  the  1st  into  tlie  square  of 
the  2d  +  the  square  of  the  2d.  Cube  of  the  1st  —  3  times  tlae 
square  of  the  1st  into  the  2d  +  3  times  the  1st  into  the  square  of 
the  2d  —  the  cube  of  the  2d.  14  4 ,/■-  +  12 xy  +  9 //-  9 x-  —  2\x 
+  16     a--  +  4.r//  +  4y     9«2  — 3G.7.7/  +  36/.        15  1 +  3./'  +  3,r^'+.r3 

8  a;''  + 12  xhj  +  6  xf  +  //'^     27  x'  —  54  x~  +  3(3-8      x""  —  9  x''y  +  27  .r// 

—  27/.  16  16x^-\-8ax  +  a'  8a-  — 36a-2  + 54.r  -  27  IG^r 
+  2Aab-]-9b'      8«^— G0«2  +  l50rt  — 125.         17  5Qxy.        18  30 ^r 

—  Aab  — 15 «-.  19  54 a'^b  + 18 ^z".  20-7 .r"  + 15 x'-  +  9 ,'•  +  2 
-{-12x-y  —  30xy'+7y'.  21  2  X  2  X  a  X  a  2X2X2Xa  XaXa 
2x2X2x2XaXaXb  2 X 2 X o X  a  X  a  X  a  X b X b X c 
(((  +  1;  (a  +  1)  X  /// .  22  a  (x  +  y)  a  (ax  —  y)  x  X  x  X  x  {a-  +  a 
+  l;(a-l)       aXa{b-x).         23   (a +  &)(«- 1)        {a-\-b){a  +  b) 


VIII.  24:]  teachers'    INIANITAL.  191 

(x  + 1)  (x  +  1)  (a-  +  1 )  (x  -  1).  24   (<'  +  ^0  (a  +  h)  (a  +  /^ 

(.^— /;)(r^2_|.,,^,^^^2^  Q,_^l,s^(^^,2_^^l^_^f'2y  25     (2  ^/.  +  3 />)  (2  rt 

+  8  b)  (•• !  .r  +  D)  (.'5  ,r  +  9)  (3  ,r  +  '.))  (;^>  .r  -  <J) .  262X2 

X  ab  (2  ^r'  —  <>/>  +  ;;  rr')      2  X  2  X  2  X  a  X(iXaXbXb(^2  —  a}r  ^1n). 
21  2(a-  +  yH2.r  +  l)     4(.r  +  2^)(2  +  y). 

10  — « 
34    Alls  ire  rs  :     1    .t^8  —  //  x  = —  x  =  12  +  2y 

x  =  2(i/  +  zy  2  .T  =  4//-2         ^.^48-o//         ^^3^5 _2^. 

o  ,  ,   ,  3./  +  2Z.  A5  —  2a         ^  ^.  ^, 

3  a'  =  8  +  /;  +  a     ir  = y. x  =  — — 4  .r  =  SA  —  a  —  2b 

b  lo 

a"  = ^ •  x=^4a  —  ,>ao.  5  .r  =  lb  —  //  y  =  lh  —  aj 

(5 

=  2+y        y/  =  ;r-2        x  =  ^^^        ^  =  28-2a;        a'-^^  +  ^ 


a; 


32  — 2v/                 32-3x  20  +  3// 

v/  =  2.r-12.  6  .'■  = ^  y  = 2 ^=7 

4;r  — 20  40  — lOy  40  — 3. r  20  — y 

2/=^^       ^=~^r^       y=~-r~      ""^-^ir 

o  r>  '''  +  1-'/  16.X'  — (/.  . 

,;  =  20-2r.  7  :^-=       j^  y=       ^^  a.  =  a-4y 

r? — .r  8y  —  6rt  +  i  —  1  Ga  —  ^  +  l+4a; 

•'^  -  ^^         -^^^  =  "^ 4 ^  ^ 8 

6                  6y                 Ix                 Zay  —  16  a 
8. .  =  -4^  v/  =  -~  .■  =  -  ,/  =  -         :«  = j^^— 

y  =  12.r  +  16r^           9  ./■  =  4  r).*'  =  20.  10  .'=5  y/  =  4. 

11  a^  =  5.        13  ./■  =  8      //  =  2.        14  .r  =  5  //  =  (;.  15  .'■  =  7 

?/  =  5.          16  x  =  o       y  =  ^.  17  a;  =  9  y  =  2.  18  .c  =  3 
i/  =  10. 

25  Anawers:  1  ?/  =  6  .r  =  8.  2  ./■  =  6  y  =  2.  3  a- =  4 
2/  =  10.  4  .r  =  3  y/  =  2.  5  .*■  =  (;  y  =  \.  6  a-  =  5  ?/  =  r). 
7  a;  =  8         y  =  3.  8  ./-^lO         y  =  %.  9  .r  =  9         ^  =  7 

10  a;  =  12      .y  =  4.         11  x  =  9      ^=12.         12  .r  =  10      y  =  Vl. 
13  a;  =  15       y  =  9.         14  a;  =  8       ^  =  8.         15  x  =  Vl       y  =  4. 


192  GRADED    ARITHMETIC.  [VIII.  26 

16  ic  =  18     y  =  8.       17  ic  =  20     y  =  24.        18  ^-  =  2^8^     y  =  —  2. 

19  a^  =  40  ?/  =  24.  20  .'k  =  30  v/  =  48.  21  ic  =  14f  3/  =  48 
s  =  26f.      22  .c  =  12     y  =  2     «  =  10.       23  x-  =  9     y  =  20     2;  =  6. 

26  Answers:   1  x==2.         2   y  =  A^.         3  y  =  6.         4  ?/  =  2. 

5  x  =  8  y  =  6.  6  x  =  9  y=10.  7  :t'  =  o  ?/  =  6.  8  .f  =  8 
y  =  3.  9  .«  =  — 3a  //  =  4f.  10  x  =  9  y  =  S.  11  a'  =  8 
2/  =  12.  12  a=7  ^  =  2.  13  .r  =  12  ?/  =  10.  14  .):  =  15 
2/  =  18.  15  a.  =  34f       ^  =  22^.  16  x  =  l{f       l/  =  -^\. 

17  x  =  6     ?/  =  16.        18  ir  =  67i     y  =  4(3^.         19  rc  =  42     y  =  3. 

20  x  =  6  i/  =  25.  21  «  =  6  .r  =  8  y  =  4  2;  =  9.  22  «  =  18 
a;  =  8     v/  =  12     «  =  6. 

27  Answers:  1  ,t  =  8  ?/  =  5.  2  a;  =  6  ?/  =  10.  3  x  =  12 
2/  =  20.  4  x  =  9  7/  =  8.  5  .r  =  12.  y  =  2.  6  x  =  4  ^  =  16. 
7  a;  =  20  ij  =  15.  8  .r  =  18  ?/  =  7.  9  x  =  7j'-g  ^  =  14^2-. 
10  ii:  =  28f  y/  =  — 14f.  11  .r  =  33  y  =  30.  12  :i'  =  12 
3/ =  10.  13  a:  =  20  y  =  18.  14  a-  =  10|  y  =  — 13if 
15  x  =  ()  v/  =  lo.  16  a' =  20  y  =  10.  17  .r  =  2l'if 
y/  =  13_j7_.  18  if  =  30  y  =  24.  19  x  =  8  y  =  12.  20  *  =  12 
?/  =  2.  21  a' =  9  y  =  4.  22  a- =  22^1-  ^/^I^tV 
23  a;  =  21|f  v/  =  20|f.  24  .r  =  12  2/ ==20.  25  .r  =  G 
y  =  8     ,^  =  16.       26  .r  =  5     y  =  30     «  =  10. 

28  Ansivers  :  1  a-\-b  —  c  a  —  h  —  c  a — b  —  c  a  —  b-\-c 
a-{'b  —  e  —  d  +  e.  2  a''  +  2ab  +  b^  d-  —  V\  3  ^a^ —  ^ab-\-2iW 
—  6/r  +  23Z'c  — 20('l       4  56"2  — r)c  +  5crf  +  13r/.       5  c  +  rf     a  —  b. 

6  a^^y        a^  —  irh^a%''—ah''-^b\  7  « +  i  — e  + ^/  — 3^/  + A. 

2  //72     I     7  2\  4      7 

8K«-1)     ^(-^-1)     «(^-2y  +  3..).         9     \.Zy.^     -^^~^,' 
10  («  +  &)'      «2+^r  +  2«Z-      («-^^)'       «'  — ^'       («  +  ^>)  X  0?  — />) 

{a-\-hf.         11—4     TTT j—      — — :•  12  6     f 

^  ^  a  12  —  n     m-\-n     a-\-\  a 

_6     ±Z1     iizi.  13  9     12     2     ^.  14  7^     - 

a  a  —  b  c  0-f-c      ac 


VIII.  29]  TEACHEKS'    MANUAL.  193 

'^'^-"^       ;^^3^,       ^^^'Uh         15  22i       5'-     ^     ^       12.247+ 

39  Jwsu-ers.-  16  8.  2  18  8.  3  20  8.  4  John,  12/ 
James,  18/.  5  Apple,  f  /  orange,  5|/.  6  6f ,  2|.  7  15  25. 
8  Man,  50  wife,  40  daughter,  10.  9  20  yr.  10  James,  18 
John,  12.  11  Coffee,  24-i<V/  tea,  65/^^/.  12  lilOO.  13  A, 
$29     B,  $11.       14  656  apples. 

30  A7isirers :  1  $480.  2  $540.  3  A,  50  B,  30.  4  36  da. 
5  2.4  h.  6  40  persons.  7  A,  $7  B,  $5.  8  Pigeon,  40/ 
chicken,  50/.  9  10  sheep,  20  calves.  10  $400.  11  Jane, 
$250     Sarah,  $400     Ellen,  $150     Mary,  $200.       12  15     8. 

9  is  indeterminate.  Other  answers  are,  5  sheep  and  27  calves  ; 
15  sheep  and  13  calves ;  20  sheep  and  6  calves. 

31  Ansivers:  1  Apple,  2/  pear,  1/  orange,  4/.  2  $36000. 
3  13^  da.  4  6i  li.  5  105}  78f.  6  $1500  @  6%.  7  8, 
12.  9  A,  9|1  da.  B,  15|i  da.  C,  8|a  da.  10  $3500  in  4's 
$1500  in  5's.       11  ^'^.       12  48. 

8  is  indeterminate,  any  one  of  forty  answers  being  correct. 

32  Ansivers :  1  Nile,  3000  miles  Danube,  1600  miles  Ama- 
zon, 3600  miles.  2  St.  Peter's,  442  ft.  Bunker  Hill,  215  ft. 
Washington,  549  ft.      3  3  h.  16/j-  min.     6  h.  32^^  min.      4  $1400 

$1600.      5  14|  h.      6  8  da.      7  6i    li-      8  ^"^^~  ■^  -      9  -^, 

(till  al  am  amo  /lOO  +  »\'" 

^;r+i'         j»+i    VI + 1    w(m+o'  "v  100  /  ■ 

ant (d  —  c)               hm  (d  —  c)  _  _  ah  -,  a    -to 

12   x  =  —r^ j^     y  =  --r^ f-'        13  J— r — mo.        14  12 

he  —  ad  he  —  ad  bm  -\-  an 

20.       15  12  h.  32^8^  min. 


194  BEADED   ARITHMETIC.  [A^III.  33 

33  Ansu-er.^:  12  192.  13  IDS.  14  729.  15  S192.  16  2^. 
17  11      18  .729.      19  .0000078125.      20  f.      21  108.      22  283. 

23  Iji^.  24  .0225.  25  .244/^.  26  ^t^Vittt-  27  ^1^. 
28  1600  7056  9801  10000  313600.  29  698900  106276 
474721  998001  1000000.  30  81796.  31  20.9764. 
32  .00005625.      33  27.210701.      34  31640625.      35  1.10271001. 

36  41371138.5616.  37  94a||.  38  32768.  39  88.121125. 
40  5861fg|.       41  383328f 

34  The  formula  for  the  extraction  of  the  square  root  of  a 
number  can  be  found  by  finding  the  square  of  the  root :  t  -\-  u. 
Pupils  who  have  taken  the  algebraic  exercises  will  have  no  diffi- 
culty in  this.  Others  will  have  to  be  taught.  If  the  symbols 
representing  any  root  be  found  too  difficult  to  work  with,  let  figures 
representing  the  root  of  a  given  power  be  used,  e.g.,  the  square  of 
25  =  20-  +  2  X  (20  X  5)  +  5^. 

35  A7mcers:  1  32.  2  43.  3  51.  4  62.  5  73.  6  35. 
7  46.  8  36.  9  57.  10  76.  11  67.  12  74.  13  83. 
14  94.      15  89.      16  07.      20  4.2.      21  6.3.      22  5.4.      23  3.5. 

24  5.6.  25  4.7.  26  5.,S.  27  6.9.  28  S.S.  29  0.6.  30  9.9. 
31  7.9.        32  7.9.        33  231.        34  342.        35  254.        36  364. 

37  286.  38  523.  39  475.  40  279.  41  634.  42  488. 
43  596.  44  678.  45  969.  46  804.  47  709.  48  232.1. 
49  354.6.  50  465.8.  51  50.76.  52  60.19.  53  1.41 +  . 
54  6.32+.  55  15.55+.  56  37.14+.  57  40.92+.  58  1.87  +  . 
59  4.02+.  60  2.69+.  61  2.84+.  62  12.66  +  .  63  5.72+. 
64  2.10+.  65  30.009+.  66  2.23+.  67  3.61 +  .  68  10.83+. 
69  5.22+.       70  .64  +  .       71  .87  +  .       72  .73+.       73  .52  +  . 

36  Answers:  1  14  rd.  2  186  ft.  274  ft.  511.23+  ft. 
3  12.649+ rd.  6.324+ rd.  4  9.8+ rd.  on  two  sides.  5  254.13+. 
6  884.84  ft.  7  148  blocks  98|  ft.  square.  8  056871  paving 
stones.  9  14.14+  in.  10  24  ft.  by  16  ft.  11  12.29  ft.  on 
longer  side  12.11  ft.  on  shorter  side.  12  154.5  rd.  13  19 
and  15.       14  1141.40.       15  $141.40. 


Till.  :J7]  TEACHEKS'    ISIANUAL.  19o 

37  Ansirers:  7  80.  8  o2.  9  41.  10  lo.  11  68.  12  63. 
13  99.  14  97.  15  4.7.  16  33.  17  12.7.  18  .89.  19  362. 
20  41o.     21  472.     22  903. 

:^H  Jiisn-ers:  1  1.58+.  2  3.91+.  3  5.64+.  4  11.69  +  . 
5  1.6.S  +  .  6  2.43+.  7  1.8!)  +  .  8  4.82+.  9  1.91+.  10  2.38+. 
11  .79+.  12  .42+.  13  .76  +  .  14  .43  +  .  15  .39+.  16  .74+. 
17  13  in.  35  in.  18  87  in.  72  in.  19  12.9+  in.  47.55+ in. 
20  73  in.  28.4+  in.  21  1.41+  ft.  22  10.06 +  rd.  by  15.09+  rd. 
23  12.14+  ft.  24  6  ft.  long,  3  ft.  wide,  1^  ft.  deep.  25  8.07+  ft. 
26  160  rd.     27  88.4  + in.     132.6  + in.     176.8  + in.     28  150.26 +  rd. 

39  The  following  facts  should  be  drawn  from  the  pupils  by 
teaching,  as  Jaefore  shown  : 

flatter  is  anything  we  get  a  knowledge  of  through  the  senses. 

A  Ikh/i/  is  a  limited  portion  of  matter. 

Space  is  the  room  a  l)ody  occupies,  and  the  room  that  is  around 
a  body. 

A  volume  is  a  limited  portion  of  space.  A  volume  may  be  repre- 
sented by  a  so/id,  wliieh  has  three  dimensions,  length,  breadth,  and 
thickness. 

A  surface  is  the  limit  of  a  volume,  and  has  only  two  dimensions, 
length  and  breadth, 

A  line  is  the  limit  of  a  surface,  and  has  only  one  dimension, 
length. 

K.  iwlnt  is  the  limit  of  a  line.  It  has  position  only.  It  can  be 
represented  by  a  dot. 

A  straight  line  is  a  line  which  has  the  same  direction  throughout 
its  entire  length. 

A  curved  line  is  a  line  that  constantly  changes  its  direction. 

(For  definitions  of  horizontal  line  and  vertical  line,  see  Manual, 
page  168.) 

Lines  are  parallel  when  they  have  the  same  direction.  However 
far  prolonged,  tliey  can  never  meet.  "When  one  line  meets  another 
line  so  as  to  make  the  adjacent  angles  equal,  the  lines  are  said 
to  be  perp>endicidar  to  each  other. 


196  GRADED  aeith:metic.  [VIII.  40 

9  A  line  may  be  drawn  parallel  to  another  line  as  previously 
shown,  or  by  the  following  way  : 

To  draw  through  the  point  It  a  line 
parallel  to  the  given  line  OF. 

From  the  point  C,  with  a  radius  equal 

to  CB,  draw  the  semi-circumference  ARB. 

From  5  as  a  center,  with  a  radius  equal 

to  AB,  cut  the  circumference  at  S.     Join  BS,  and  Ave  have  a  line 

parallel  to  OP.     (Let  the  pupils  show  why  the  lines  are  parallel.) 

11-12    For  a  method  of  dividing  a  line  into  2,  4,  or  8  equal 

parts,  see  Manual,  page  169.     The  following  method  of  dividing 

Q  a  line  into  any  number  of 

"'~~--.,^^  f>  parts  may  be  ttiught : 

/     ~-~-~-,       ^  To  divide  a  line  AB  into 

/  ?'~~*--.^  equal  parts. 

/  """7-^.  Draw  the  line  AO  ot  any 

^/ ^ / '''---..^  ^    length,  and  lay  off  on  that 

■B  ^  ^  line  parts  of  any  conven- 

ient equal  length.  Join  BC,  and  through  the  points  of  division 
on  ^0  draw  lines  parallel  to  BC.  These  lines  divide  AB  into 
equal  parts. 

The  standard  unit  of  length  in  this  country  is  the  yard,  the 
same  as  the  imperial  yard  of  Great  Britain.  Its  length  is  §f  y§§§ 
of  the  length  of  a  pendulum  which  vibrates  seconds  in  a  vacuum 
at  the  level  of  the  sea  at  62°  Fahrenheit  in  the  latitude  of  London. 
Other  interesting  facts  concerning  how  and  where  the  standard 
yard  is  kept  can  be  gathered  from  a  cyclopedia  and  given  to  the 
pupils. 

40   Teach,  as  before,  the  following  definitions  : 

An  anf/le  is  the  difference  of  direction  of  two  lines  in  the  same 
jdane.  The  i)oint  where  the  two  lines  meet  is  the  vertex  of  the 
angle.  A  7'if/ht  (nu/le  is  the  difference  of  direction  half  as  great 
as  o])]iositeness  ;  or,  it  is  an  angle  formed  by  two  lines  extending 
perpendicularly   from    each    other.     An    obtuse  angle  is   an  angle 


VIII.  41] 


TEACHERS     MANUAL. 


197 


greater  than  a  right  angle.  An  acute  angle  is  an  angle  less  than 
a  right  angle. 

An  angle  is  measured  on  the  circumference  of  the  circle  whose 
center  is  at  the  vertex  of  the  angle.  The  unit  of  measui:e  is  a 
degree,  which  is  ^^^  of  the  circumference  of  a  circle.  The  follow- 
ing figures  will  suggest  a  method  of  making  an  angle  equal  to  a 
given  angle,  and  also  for  making  angles  equal  to  twice  or  three 
times  the  size  of  a  given  angle  : 

13    To  draAV  an  angle  equal  to  a  given  angle. 


The  pupils  wall  see  that  the  proof  of  the  equality  of  these  angles 
rests  upon  the  fact  that  equal  arcs  subtend  equal  angles.  This  is 
implied  in  Avhat  they  have  learned  about  the  measurement  of 
angles. 

14  To  draw  an  angle  twice  and  three  times  the  size  of  a  given 
angle. 


In  teaching  the  above  problems,  as  well  as  all  that  follow,  the 
teacher  should  give  as  little  direct  assistance  as  possible,  leading 
them  on  slowly  by  suggestions  and  questions. 

41  Vertical  angles  are  angles  that  have  a  common  vertex,  and 
that  have  sides  extending  in  opposite  directions. 

Adjacent  angles  (a  and  h  in  the  figiire)  are  angles  that  have  the 
vertex  and  one  side  common,  and  Avhose  other  sides  are  opposite 
parts  of  the  same  straight  line.  (Tt  will  be  observed  that  one 
element  of  adjacent  angles  is  left  out  in  2.) 


198 


GKADED   ARITHMETIC, 


[A  III.  42 


4   To  prove  that  the  sum  of  two  adjacent  angles  is  equal  to  two 
right  angles. 

Draw  DO  perpendicular  to  AB. 
AOC  +  BOC  =  ADD  +  BOD. 
A0D+B0D  =  2  right  angles. 
-r     .•.A0C'i-B0C=2  right  angles. 


B 


0 


In  7,  lead  the  pupils  to  see  and  to  say  tliat  the  angles  a  and  b 
(in  the  figure,  lOj  =  the  angles  b  and  c.  Taking  away  the  com- 
mon angle  b,  the  angle  a  =  the  angle  c. 

In  10,  the  four  angles  c,  d,  n,  and  m  are  called  internal  angles, 
because  they  lie  between  the  parallel  lines  ;  and  the  four  angles 
a,  b,  0,  and  j)  are  called  external  angles.  The  angles  a  and  m  and 
the  angles  b  and  7i  are  exterior-interior  angles.  The  angles  c 
and  m  and  the  angles  d  and  n  are  alter^iate-interior  angles.  Lead 
the  pupils  to  discover  what  angles  are  equal,  and  why  they  are 
equal.     The  lines  AB  and  CD  are  drawn  parallel  to  each  other. 

A.  plane  surface  is  such  a  surface  that,  if  any  two  points  in  it 
be  connected  by  a  straiglit  line,  tliat  line  will  lie  Avholly  in  the 
surface. 

Let  the  definitions  called  for  in  12  be  concise  and  comprehensive. 
(For  method  of  teaching,  see  Manual,  pages  104  and  116.) 

43  Let  the  solution  of  the  theorems  and  problems  be  made  by 
measurement  and  construction,  and  also  by  demonstrations  so  far 
as  the  pupils  can  be  led  to  make  them  or  to  understand  them. 
The  following  figures  and  liints  may  suggest  demonstrations  of  the 
more  difficult  propositions  : 


3    Draw  CE  II  to  AB. 
Prolong  J  C  to  F. 
EOF  +  ECB  +  ACB  =  2 
angles. 

A  =  E(  'F. 
B  =  EC 'J. 


right 


VIII.  48] 


TEACHERS     ]\LA.NUAL. 


199 


Let  tlie  order  of  demonstrating  5,  6,  and  7  be  reversed. 


6   Bisect  angle  C. 
AC=CB. 
CD  common. 

.'.  a  =  b. 


From  the  fact  proved  in  6,  it  can  be  shown  that  the  angles  of 
an  equilateral  triangle  are  equal. 


AC=  CB. 

AD  =  DB. 

Join  AB. 
CAB  =  ABC. 
BAD  =  ABD. 
CAD  =  CBD. 


The  exercises  from  11  to  15  are  to  be  performed  with  protractor 
and  ruler. 

43    The  pupils  may  be  able  to  solve  3  by  demonstration. 

In  4,  lead  the  pupils  to  prove  that  the  two  triangles  are  equal, 
that  the  angle  at  the  vertex  is  bisected  by  the 
line   drawn   from  the  middle  point  of  base  to 
vertex,  and  that  the  base  is  bisected. 

This  figure  Avill  suggest  a  method  of  bisecting 
the  angle  A.  Lead  the  pupils  to  prove  that 
the  triangle  Jlfi)  =  the  triangle  AND,  and 
that  therefore  the  angle  A  is  bisected. 

For  a  method  of  bisecting  a  line,  see  Manual, 
page  169, 


200 


GRADED    ARITHMETIC. 


[VIII.  44 


The  following  figures  will  suggest  ways  of  solving  the  problems 
contained  in  7  : 


Ey, 


c 


■'R 


\ 


F 


H 


.B 


B 


It  would  be  well  to  apply  the  principles  involved  in  the  last 
exercises  of  the  page  by  actually  measuring  distances  in  a  field. 
This  may  be  done  with  lines  and  stakes  or  with  stakes  alone. 

4:4  The  definitions  called  for  on  this  page  are  supposed  to  have 
been  taught  previously  (see  Manual,  page  104). 

The  simplest  solution  of  3  may  be  made  by  dividing  a  quadri- 
lateral into  2  triangles. 

The  propositions  on  this  page  can  all  be  performed  by  measure- 
ment or  construction.  As  many  of  them  should  be  demonstrated 
by  the   pupils  as  possible.     They  will  not  be  found  difficult  for 

those  who  have  demonstrated  the 
preceding  propositions.  The  fol- 
lowing hints  might  be  given  if 
necessary  : 

What  angles  do  you  know  to 
be  equal  ?  AVhat  lines  ?  Compare 
the  size  of  the  two  triangles.  If 
we  were  to  prolong  one  of  the  sides,  what  two  angles  are  equal 
to  2  right  angles  ?  What  other  two  angles  must  be  equal  to  2 
right  angles  ? 


li  :r- 


What  angles  can  you  prove  to  be  equal  ? 
Wliat  lines?  What  triangles?  What 
other  lines  ? 


''E 


B 


VIII.  45]  teachers'  manual.  201 

The  pupils  may  be  told  that  an  isosceles  trapezoid  is  a  trapezoid 
whose  sides  between  the  two  parallel  lines  are  equal.  TJie  propo- 
sitions contained  in  13  to  16  can  be  easily  proved. 

45  A  polygon  with  two  reentrant  angles  : 
Exercises  may  be  given  to  show  how  the  area 

of  such  a  polygon  may  be  found.  Let  the  pupils 
discover  two  ways,  and  what  dimensions  must 
be  known. 

The  simplest  way  to  construct  a  regular  poly- 
gon is  from  the  circle.  Show  to  the  pupils  that  by  means  of  com- 
passes the  circumference  of  a  circle  can  be  divided  into  any  number 
of  equal  parts,  and  that  the  straight  lines  connecting  the  points  of 
division  are  the  sides  of  a  regular  polygon. 

The  formulas  called  for  in  15  and  16  are,  S=ah;  a  =  S'T--x', 

46  The  triangles  ADE  and  ABC  may  be  called  similar,  because 
they  are  of  the  same  shape.  Later  (page  50)  the  pupils  will  learn 
more  definitely  what  similar  polygons  are.  Triangles  are  equiva- 
lent if  they  have  the  same  size,  and  equal  if  they  have  the  same 
shape  and  size. 

The  following  formulas  should  be  made  by  the  pupils  from  their 
knowledge  of  finding  areas  : 

Triangle,  S  =  ~]  traipezoid, — ^r-^XA;  regular  polygon,  8^=—' 

The  transformation  of  polygons  into  equivalent  polygons  of  any 
required  shape  can  be  readily  made  if  the  method  of  finding  the 
areas  is  thoroughly  understood. 

47  Answers:  1  i ooo  a.  2  1728  sq.  ft.  4290  sq.  ft.  252T8f 
sq.  ft.  3  272i  ft.  4  80  sq.  ft.  2016  sq.  ft.  5  1512  sq.  ft. 
61640  sq.  ft.  6  217|  ft.  7  147.5+  ft.  8  153  sq.  ft.  9  3  ft. 
8.3+  in.  10  344.6+  ft.  11  141.7+  ft.  12  4380  sq.ft.  13  -}. 
14  17+  bundles. 

48  A7iswers:  5  5  in.    15  in.     6  8  in.    13.41+  ft.     7  25.98+ yd. 


202  GRADED    ARITHMETIC.  [VIII.  49 

The    pupils    should   be    encouraged   to   give   other  proofs   than 
those   here   given.     The  formulas  to  be  given  are  :  h  =  V*^^  +  j^"^ ; 


in 


49  Ansii-ers:  1  100  ft.  2  25  ft.  3  43.8+  rd.  4  60 
5  62.4+  ft.  6  120  rd.  7  19.59+  yd.  11  34.66+  ft.  693.2+ 
sq.  ft.  12  25.45+  ft.  13  70.71+  ft.  14  24.08+  ft.  32.8+  ft. 
15  45.69+  ft.  16  30.98+  ft.  17  36.76+  ft.  18  65.28+  rd. 
19  25.45+  rd.       20  26.07+  ft.       21  722.9+  ft. 

50  The  term  covresponding  may  be  used  instead  of  homologous, 
if  preferred.  The  proof  called  for  in  6  may  be  experimental  rather 
than  demonstrative.  It  follows  what  is  supposed  to  be  done  in 
previous  exercises. 

The  proof  called  for  in  7  may  be  made  from  measurement  and 
comparison  of  the  sides  of  similar  polygons. 

51  Answers:  1  9  times  as  large.  24:1.  32:1.  45:1 
25  :  1.  5  438f  sq.  ft.  6  94.8+  ft.  7  5250  sq.  rd.  8  222-| 
sq.  ft.       9  242.4+  ft.       10  2.4+  ft.  .    11  35f  ft.       12  36  ft. 

These  exercises  should  be  performed  by  proportion.     It  would 

be  well  for  pupils  to  write  the  proportions  and  statements  in  full ; 

thus,  in  5  : 

(40  ft.)2 :  (50  ft.)2  =  300  sq.  ft.  :  number  of  sq.  ft.  in  larger  triangle. 

300  X  2500  .         „         „     .     ,  ^  .       , 

■ ■=^  number  of  sq.  ft.  m  larger  triangle. 

and  in  6  : 

4000  sq.  ft.  :  10000  sq.  ft.  =  60^  :  the  square  of  the  side  of  larger 

hexagon. 

10000  X  3600  .  ,1       . .       n  1  . 

-  =  the  square  of  the  side  of  larger  hexagon. 


4000 


4 


10000X3600        .-,      ^, 

-  =  side  of  larc:er  hexagon. 


4000 


LC,V.X       iiV^^lVj^V. 


52   Answers :  1  33  it.     2  93^  ft.     3  27f.    4  29T\ft.    6  86 §  ft, 
7  43^  ft. 


VIII.  r>3]  teachers'  manual.  203 

In  8,  a  line  may  be  tlraAvn  downward  from  7>  parallel  to  AX, 
and  one  from  A  parallel  to  BX,  so  as  to  meet  the  first  line  at  K 
Thus  Avould  be  found  two  equal  triangles,  two  angles  and  included 
side  of  one  triangle  being  equal  to  two  angles  and  included  side  of 
the  other. 

In  9,  extend  the  line  XI  to  a  point  F.  Draw  FG  perpendicular 
to  FX.  Join  GX.  From  the  point  A  draw  a  line  .47/"  parallel  to 
GX.  Thus  are  formed  two  similar  right-angled  triangles,  and 
FIT  :  FG  =  FA  :  FX. 

53  In  1,  the  lines  ^LY  and  ^-iFare  found  by  the  method  shown 
in  5,  page  52.  Ax  is  the  same  fractional  part  of  AX  that  Ai/  is 
of  A  Y.  The  two  triangles  Axy  and  AXY  are  similar,  and  Ax  :  AX 
=  xy  :  XY. 

The  following  definitions  may  be  taught  as  previously  shown  : 
A  circle  is  a  plane  figure  bounded  by  a  curved  line,  all  points  of 
which  are  equally  distant  from  a  point  within,  called  the  center. 
The  circuiiifeyence  of  a  circle  is  the  line  which  bounds  it.  The 
diameter  is  a  straight  line  passing  through  the  center  and  terminat- 
ing at  the  circumference.  The  radius  is  a  straight  line  connecting 
the  center  Avith  any  point  in  the  circumference.  An  arc  is  any  por- 
tion of  a  circumference.  A  chord  is  a  straight  line  connecting  the 
extremities  of  an  arc.  A  segment  is  a  portion  of  a  circle  bounded 
by  a  chord  and  its  arc.  A  sector  is  a  portion  of  a  circle  bounded 
by  two  radii  and  the  included  arc.  A  semi-circle  is  a  portion  of  a 
circle  bounded  by  a  diameter  and  half  the  circumference. 

Draw  lines  as  indicated  in  10,  and  show  that  every  point  of  the 
perpendicular  is  equidistant  from  the  ends  of  the  chord.  11  and 
12  can  be  readily  shown  from  10. 

54  In  1,  join  the  points  by  lines,  which  may  be  regarded  as 
chords  of  the  required  circle,  and  erect  perpendiculars  from  the 
middle  points. 

In  drawing  an  inscribed  angle  (4),  there  are  three  possible  con- 
ditions. The  center  may  be  in  one  of  the  sides  of  the  angle,  or, 
between  the  sides  of  the  angle,  or,  without  the  sides  of  the  angle. 
The  first  case  should  be  proved  first,  the  proof  resting  upon  the 


204  GEADED  ARITHMETIC.  [VIII.  54 

facts,  (1)  that  the  angle  i?OC  is  equal  to  the  sum  of  the  angles 

A  and  C ;  and  (2)  that  the  angles  A  and  C 
are  equal.  From  this  proof  the  other  cases 
may  be  easily  proved. 

A  secant  is  a  straight  line  cutting  the  cir- 
cumference of  a  circle  in  two  points. 

A  tangent  is  a  line  which  touches  a  circum- 
ference in  one  point  without  cutting  it. 

The  proof  of  6  rests  upon  the  facts,  (1)  that 
a  line  from  the  point  of  contact  to  the  center  is  the  shortest  dis- 
tance between  the  tangent  and  the  center  ;  and  (2)  that  the  shortest 
line  between  a  point  and  a  straight  line  is  the  perpendicular  line. 

In  8,  connect  the  point  with  the  center  of  the  circle,  and,  upon 
this  line  as  a  diameter,  describe  a  circle  which  will  cut  the  circum- 
ference of  the  first  circle  at  two  points.  Join  these  two  points 
with  the  given  point  outside  of  the  circumference,  and  the  lines 
thus  drawn  are  the  tangents  required.  This  can  be  easily  proved 
from  what  has  been  proved  before. 

Before  giving  9  to  12,  show  that  a  circle  is  inscribed  in  a  poly- 
gon when  its  circumference  touches  every  side  of  the  polygon,  and 
that  a  circle  is  circumscribed  about  a  polygon  when  its  circum- 
ference passes  through  all  the  corners  of  the  polygon. 

In  9,  the  point  of  intersection  of  the  lines  bisecting  the  angles 
is  the  center  of  the  inscribed  circle.  To  prove  that  the  sides  of 
the  triangle  are  tangents  to  the  circle,  draw  lines  from  the  center 
perpendicular  to  the  sides  of  the  triangle,  and  show  from  the 
equivalence  of  the  triangles  that  the  perpendicular  lines  are  equal, 
and  are  therefore  the  radii  of  tlie  circle  touching  all  sides  of  the 
triangle. 

In  10,  draw  such  perpendiculars  to  radii  as  will  intersect  and 
form  a  triangle. 

11  and  12  can  be  readily  done  when  it  is  understood  that  the 
bisectors  of  tlie  angles  of  a  regular  polygon  meet  in  a  point  equally 
distant  from  all  the  sides  and  all  the  corners.  This  can  be  proved 
from  the  equality  of  the  triangles. 


VIII.  56]  teachers'  manual.  205 

In  15,  the  pupils  can  readily  find  the  approximate  ratio  by 
measurement.  They  can  also  estimate  by  calculation  the  perimeter 
of  a  regular  polygon  inscribed  in  a  circle  liaving  a  given  radius, 
and  can  see  that  the  greater  the  number  of  sides  of  the  inscribed 
polygon,  the  nearer  the  perimeter  approaches  the  length  of  the 
circumference. 

55  From  what  has  been  done  previously  (see  Book  VI.,  page  18) 
the  formulas  in  1  to  5  can  be  illustrated. 

6  The  sector  COD  may  be  divided  into 
any  number  of  triangles  whose  altitude  is 
EG  and  the  sum  of  whose  bases  is  CD. 

7  The  area  of  the  segment  CED  is  equal 

to  the  sector   COD  —  the  triangle  COD,  i.e. 

^T-,T.      EO      ^_,       FO 
CED  X  -^ CD  X  — • 

To  find  the  area  of  a  circular  zone  or  cir- 
E  cular   ring,  subtract  from   the    area   of   the 

circle  the  part  not  included  in  the  zone  or  ring.  Let  the  pupils 
determine  Avhat  dimensions  must  be  given. 

Let  the  proofs  called  for  in  10  to  13  be  made  from  construction 
and  comparison  of  measurements.  Pupils  may  be  led  to  use  what 
they  already  know  of  similar  polygons. 

To  draw  an  ellipse,  stick  two  pins  or  tacks  into  the  surface  upon 
which  the  ellipse  is  to  be  drawn,  and  tie  to  them  a  string  longer 
than  the  distance  between  them.  Describe  with  a  pencil  the  curve, 
keeping  the  string  constantly  stretched  to  its  full  length.  Th(^ 
points  A  and  D  in  the  figure  are  the  foci,  CD  the  major  axis,  A]> 
the  7ni7ioi'  axis.  The  eccentricity  is  the  ratio  that  the  minor  axis 
bears  to  the  major  axis.  An  ellipse  is  a  plane  figure  bounded  by 
such  a  curved  line  that  if  from  any  point  in  it  straight  lines  be 
drawn  to  two  points  within,  called  the  foci,  their  sum  will  be  a 
constant  quantity. 

56  Ansivers :  9  175  sq.  in.  124  cu.  in.  10  600  sq.  in.,  1000 
cu.  in.  253|  sq.  ft.,  274f  cu.  ft.  80f  sq.  ft,  49^^  cu.  ft.  114ifin. 
12  4.4+  ft.     18  384  sq.  in.     19  23.34+  cu.  ft.     20  493^  cu.  in. 


206  GEADED   ARITHMETIC.  [A'III.  57 

This  and  the  following  page  of  exercises  should  be  taught  largely 
from  the  blocks.  The  following  notes  apply  to  some  of  the  more 
difficult  points  : 

A  dihedral  angle  is  the  opening  between  two  intersecting  planes. 
A  trihedral  angle  is  the  opening  of  three  planes  which  meet  at 
a  common  point. 

A  tetrahedron  is  a  solid  bounded  by  four  planes.  A  regular  pohj- 
hedron  is  a  polyhedron  having  equal  and  regular  faces  and  equal 
polyhedral  angles.  A  prism  is  a  polyhedron  bounded  by  parallelo- 
grams and  two  equal  and  parallel  polygons,  called  bases.  A  right 
jorism  is  a  prism  whose  lateral  edges  are  perpendicular  to  the  bases. 
A  parallelopiped  is  a  prism  all  of  whose  faces  are  parallelograms. 
A  pyramid  is  a  polyhedron  bounded  by  triangles  that  have  a  com- 
mon vertex,  and  by  a  polygon,  called  the  base.  A  regular  pTjramid 
is  a  pyramid  whose  base  is  a  regular  polygon,  and  whose  vertex  is 
directly  above  the  center  of  the  base.  A  quadrangular  pyramid 
is  a  pyramid  whose  base  is  a  quadrilateral.  For  other  suggestions 
relating  to  pyramids,  etc.,  see  Manual,  page  171,  and  for  rules  called 
for,  see  Appendix,  page  115. 

57  Answers:  11  207.3456  sq.  ft.  226.1952  cu.  ft.  502.656 
sq.  in.  4021.248  cu.  in.  12  10.472  sq.  ft.  28.4925+  sq.  ft. 
13  3.5342  cu.  ft.  41.4517  cu.  ft.  14  2869.0858  cu.  in.  15  804.2496 
sq.  in.  2144.6656  cu.  in.  226.9818  sq.  ft.,  321.5575  cu.  ft.  2576.896 
sq.  ft.,  10761.79392  cu.  ft.  16  25.4+ in.  17  6  sq.  ft.  2.76+ ft. 
8.676+  ft.       18  10.4+  ft.       19  2.01+  in.       20  32.127+  gab 

A  sphere  is  a  volume  that  may  be  generated  by  the  rotation  of 
a  semi-circle  upon  the  diameter  as  an  axis.  The  diameter  of  a 
sphere  is  a  straight  line  passing  through  the  center  and  having 
its  extremities  in  the  surface.  A  great  circle  of  a  sphere  is  a 
section  made  by  cutting  the  sphere  through  the  center.  A  small 
circle  of  a  sphere  is  a  section  made  by  cutting  the  sphere  outside 
the  center.  A  spherical  zone  is  the  surface  of  a  sphere  included 
between  the  circumferences  of  two  parallel  circles,  or  between  the 
circumference  of  a  circle  and  a  parallel  plane  tangent  to  the  sphere. 


VIIT.  58]  teachers'   manual.  ^0? 

A  spherical  segment  is  a  portion  of  a  sphere  included  between  two 
parallel  circles,  or  between  a  circle  and  a  parallel  plane  tangent 
to  the  sphere.  A  spherical  sector  is  the  portion  of  a  sphere  which 
may  be  generated  by  the  rotation  of  a  sector  of  a  circle  upon  a 
diameter  of  the  sphere  as  an  axis. 

Notes  applying  to  other  exercises  on  this  page  will  be  found  on 
pages  172  and  173  of  the  Manual. 

58  Anstvers:   5  499.2  lb.       1684.8  lb.       1.11+  ft.      2.52+  ft. 

6  8:1.  7  5  ft.  8  2  ft.  9  27  cubes.  10  12.7  ft.  11  118|ibu. 
G  ft.  3.4+  in.       12  13.3+  ft.       13  4.3+  ft.  X  6.4+  ft. 

59  Ansivers:  1  10.122+  ft.  15.69+  ft.  12.4+  ft.  10.5+  ft. 
2  («)  80  ft.;  {h)  1432+eu.  ft.;  {<■)  14.9+ cords.  3  2010624000000 
sq.  mi.  25673657856000000  sq.  mi.  4  9  times  27  times.  5  Moon 
=  .001801824  of  earth     Earth  =  i^^\^^^  of  the  sun.       6  9.4+  in. 

7  Surface,  432  sq.  in.  Volume,  610.56+  cu.  in.  8  3.04+  in. 
2.+ in.,  2.5+ in.,  10.39+ in.     9  Diameter,  11.6+ in.    Depth,  5.6+in. 

10  624  balls.      11  3040+ cu.  in.      12  6.075  T.      13  17.7  +  ™™ 

GO    Ansicers:    1   (r/)   12  ft.;    (Ji)   12   ft.;    (c)   8*  ft.;    ((/)   llyV  ft- 

2  {a)  13ift.;  {b)  12i|;  {<■)  29^1  ft.  3  )i!41.S5.  4  4819.5  ft. 
5  $27.18.  6  10^  bundles  $2.83^  (allowing  h  of  openings). 
7  58  bundles.  8  (<i)  2401f  ft.;  {h)  2202i  ft.;  (^•)  1908  ft.; 
{(1)  $74.78;   (e)  $193.36. 

61    Ansivers:    1    450    sq.    ft.       2700    sq.    ft.         2    2i|§a    C. 

3  14-/U)j  C.  4  58fg.  5  $42.  6  113  cd.  7  7104  bricks 
$99.46.       8  195840  bricks     172836  bricks.       9  36.35  gal. 

63    Ansicers:    1    {a)  109494    bricks;  {h)   $1067.57;    (c)  130 

bundles;  {<!)  8816  sq.  ft.       2  {a)  $8.17;  {b)  $6.48;  {(■)  $15.44; 

{d)  18  rolls;  (e)  39 J^  yd.  3  $18.50.  4  311  yd.  29||  yd. 
5  15  rolls     $23.94     $5.30. 

63    Answers:   1  About  15  T.     2  2.15+T.     3  5.4+T.  4  10.9+ 

ft.       5  9iaft.       6  422.4  cd.     1 267.2  bu.     422.4  cu.  yd.  7  7480 

gal.        8  97.9+  bl)l.        9  154.9+  lb.        10  About  42  C.  $297.60. 

11  697.5  T.     3555|  sq.  yd.       12  $24.92. 


208  GEADED   ARITHMETIC.  [VIII.  64 

64  Answers:   2  38| bcl.  ft.      3  82^ ft.      4  935ft.      5  37|ift. 

65  A7isivers:  1  57  sq.  ft.  $4.50  47|  sq.  ft.  2938  packages. 
2  848  boxes.       3  41.5  sq.  ft.     $11.14. 

66  Ansivers:   2  63+ ft.     10^     8     49.6+ sq.  ft.     8/. 

67  Ansivers  :  1  14.18+  sq.  ft.  60.13+  sq.  ft.  2  50.26+  sq.  in. 
201.06+  sq.  rd.  3  14.28+  rd.  25.72+  rd.  4  19.635  sq.  ft. 
10.90+  sq.  ft.    4.36+  sq.  ft.    17.45+  sq.  ft.       5  80  rd.      6  $70.20. 

7  5392  ft.  8  60  yd.  58.7+yd.  9  Each  side,  35.3+ft.  10  Length 
of  parts,  96  ft.  and  54  ft.  Area,  1944  sq.  ft.  and  3456  sq.  ft. 
11  1413.72  sq.  ft.       12  $1140.02.       13  31.8  lb.       14  51.9  lb. 

68  Answers :  1  60  ft.  2  2400  sq.  yd.  3  1692  sq.  ft.  4  Each 
side,  50  yd.  Altitude,  48  yd.  5  Surfaces,  1  to  4  ;  volumes,  1 
to  8.  6  if .  7  2  in.  8  8  times  as  much.  9  $5.18.  10  25.1328 
in.  11  58.8+  ft.  12  157.08  sq.  ft.  13  17.7+  ft.  14  42.423+ 
ft.  15  Triangle,  996  sq.  ft. ;  square,  1296  sq.  ft. ;  circle,  1574.48 
sq.  ft.       16  $90.80.       17  Volume  of  frustum,  fj  of  cone. 

69  Ansu-ers:  1  5026.56  sq.  ft.  11309.76  sq.  ft.  485.43+ sq. 
in.  67.5+  sq.  ft.  2  7.0+  ft.  3  2901.6+  sq.  ft.  4  1890.7+  ft. 
5  4.9+  tons.  6  12.4+  ft.  X  38.3+  in.  X  22.1+  in.  7  794596 
sq.  ft.       8  12  shingles     8  shingles     64+  bundles.       9  8.485  rd. 

10  1022.9+'".       11  64  rd.  6  ft.       12  10  mi.     3  hr.  32  min. 

70  A7iswers:  1  65.45  cu.  in.  2  20.6+ .  3  1.49+  sq.  in. 
4  196.7  ft.        5  170.3  ft.        6  $44.44.        7  51.6  ft.      24510  ft. 

8  3§f   ft.        9  15.5+  in.         10  460.1  ft.      4  A.  135jy^-  sq.  rd. 

11  2261.95+  sq.  ft.     537.21+  sq.  ft. 

71-73  Nearly  all  these  exercises  are  a  review  of  business 
exercises  given  in  Books  VI.  and  VII.  Lead  the  pupils  to  give 
concrete  examples  before  definitions.  In  some  cases,  as  in  describ- 
ing the  various  features  of  a  promissory  note,  it  Avould  be  well  to 
have  the  examples  written  out  in  full. 

In  some  States  no  days  of  grace  are  allowed,  and  in  some  States 
days  of  grace  are  allowed  only  under  certain  circumstances.     The 


VITT.  74:-75]  TEACHERS*   MANtJAL.  209 

teacher  should  ascertain  the  hiw  and  practice  in  this  regard,  and 
explain  to  the  pupils. 

A  collateral  note  is  one  given  with  stocks,  bonds,  or  other  property 
as  security,  empowering  the  payee  to  sell  the  same  if  the  note  is 
not  paid  when  it  becomes  due.  An  accommodation  note  is  one  for 
which  tlie  maker  receives  no  consideration.  It  is  given  for  the 
purpose  of  lending  credit  to  the  payee.  Forms  of  these  notes  can 
be  obtained  at  a  bank. 

The  endorser  of  a  note  makes  himself  responsible  for  its  pay- 
ment, unless  he  writes  the  words  "  without  recourse  "  before  his 
name. 

In  answer  to  7,  page  73,  there  may  be  given  examples  of  general 
partnership,  in  which  the  partners  have  the  same  or  different 
capital,  and  examples  of  limited  partnership,  in  which  the  respon- 
sibility of  one  or  more  of  the  partners  is  limited  to  the  amount 
invested.  The  general  custom  of  merchants  as  well  as  laws  of  the 
State  regulating  partnerships  should  be  ascertained  and  explained 
to  the  pupils. 

74—75  The  rulings  of  the  cash  account  should  be  made  as 
indicated.  After  the  form  given  on  page  75  is  carefully  looked 
over  by  the  pupils,  they  may  be  asked  to  write  out  the  account  in 
full.     The  balance  Sept.  21  is  $246.61. 

76—78  These  items  should  be  used  in  writing  cash  and  per- 
sonal accounts,  as  previously  shown.  Lead  the  pupils  by  abundant 
examples  to  learn  the  use  of  the  terms  debit  and  credit,  debtor  and 
creditor.  A  person  who  receives  may  be  called  a  debtor,  and  one 
wlio  gives,  a  creditor.  It  may  be  helpful  for  pupils  in  determining 
which  side  of  the  cash  account  certain  transactions  shall  be  placed 
to  apply  the  same  distinction  of  receiving  and  giving  to  the  money- 
drawer  or  cash-box.  What  is  in  the  box  in  opening  the  account 
and  what  is  put  into  it  are  to  be  debited  to  the  account ;  what  is 
taken  out  of  the  box  and  what  remains  in  balancing  the  account 
are  to  be  credited. 

The  items  given  on  page  76  should  be  written  out  by  the  pupils 
in  the  following  form  : 


210 


GtiADEt)  Arithmetic. 


[VIII.  70 


^ 


d     <S    Its    1^    "^    ^ 

Ci   OS  ^   Qi   to   Ci 


ho 


1^ 


d ,  Co     00     <3-J     <i!     00 
to     ^  '-(     ^     SO 


i^ 


to     to 

'--t   to 


to    ^ 

to   "sj 


so 


'^    so 
00      -to 


"^ 


to 
to 


to 

o 

to 

a 

©•J 

-^o 

•'!>^ 

O 

~> 

-o 

o 

o 

s. 

, 

S 

-« 

w 

so 

e 

o 

l-< 

•cO 

s 

'■-^i 

fO 

>J 


O 

O 

•■^    *^  ^ 

~    o  S 

»0    so  S 

»  5i  -2 


K)   - 


CO 


C}   -    "^   e^  <?5   ^> 

1-^     »-H 


& 
^ 


Co     to    '^ 
*-H     <5^     ^ 


to  "^  ^  to  to 
to   J^  Co   to   to 

to 
to 

to 

U5     to 

^^  to 

to    to   to 
i^    to    Co 

so    to    to    to 
to    Co   lo   So 

to    v^  lo    "^ 
to   li^l   ^  to 

to 

to 

s<(   >~i   00   to   lo 
to               "o   '-s 

■^ 

to 

CO 

to 

'-H     t^ 

So     *~<     '3'} 

Co     So     to 
So     S>J     >-i 

'-H 

1^ 

to 

to 

■5* 

Cc    '^5    ^   to 
iti   So   Q-i   so 

(5i 

!5> 

Co 

to 
to 

■^5 

so 

so 

2.08 
1.60 

v^  lO    so 

So    'Jj    »-i 

to    Co    Co    v^ 

l~S     1-^     >^     13.J 

O 

s 

0:5 


lli 

^ 

c 

0 

c 

-^J 

^> 

■iJ 

J 

-^^ 

so 

to 

•*o 

5-; 

■u 

^ 

^ 

^ 

'c; 

^ 

»o 

f^ 

rCJ 

^0 

^ 

^ 

'^ 

'~i 

^. 

y 

CO 

w 

1^ 

h— 

« 


Oi 


.0 


lO    s-^    '^    ^O 


^    IC)     to 


to      to 


So   '^   ^   "^   So    ^  to 


to 

•«;   s 

"S    to 

^  to  tiT  ^ 

05      S  so  ,^ 

S        O  J,  rO      - 

B        .  S  -=> 

5J     SO  '~^  ^ 

^  B      ^^ 

ifi  -1:  00  "5-1  Co 


o 


^r 

00 

c«" 

oc" 

00 
to 

•^^ 

UJ 

to 

•v-j 

s^ 

r^ 

■u 

so 

rO 

t^ 

(a 

« 

<u 

H 

, 

0 
so 

S~ 

to 

, 

iiT 

h^ 

■♦0 

GO 

•0 

00 

(^ 

qT 

0 

0 

50 

<;> 

0 

s 

^^ 

'— ^ 

?i 

rO 

N 

s 

r-; 

so 

« 

fCJ 

s 

1-^ 

to 

fO 

CM 

0 

— -1 

'S 

rO 

<;^ 

<^M 

SO 

to 

so 

^*J 

00 

0 

0 

bn 

- 

- 

- 

" 

- 

- 

- 

t-^ 

1-1^::    to::    ::    "^"    - 


to    '-i    So 


^t5  to  to  ::    -    - 

>^      »-j      ■r^ 


so 

so 


VIII.  77] 


TEACHERS     MANUAL. 


211 


L- 

m 

bo 

eS 

O 

> 

■  i-l 

bjo 

a 

-f-H 

o 
ft 

<n 


<v 
be 
cj 
ft 

d 

-rH 

O 
1—1 
I— t 

o 

a> 

d 


-1-3 

o 

o 
t> 
a 
d 

<D 

rd 

H 


G 


Co   d  >--J   ^   ^  ^ 

Cio    C    b»    U5    ifj    J^ 


<u 

ii. 

o 

aj 

1. 

iJ 

^ 

<i> 

=c 

o 

lu 

a. 

CO 

o 

■*^ 

o 

2 

s 

-Si 

5 
-i; 

"55 

-o 

!~ 

<?*^ 

t-; 

« 

Oti 

t^i 

'^ 

-o 

f-H 

-o 

>-H      I^C!      Oi 
»-(      '^      ^ 


C/2 


<2)  "o  >o 

i;i   <j^  <5* 

■30 

S-      • 

Ci    i!^    >-i 

15* 

<3^ 

•^1              <2>     *~l     ^     <2)     <5^     '"I     ?^     >-l     <^1     Co     ^}     e-J 

•^ 

■=0 


s 


^ 

s~ 

<U 

00" 

-« 

s. 

«.) 

'« 

0 

.0 

5 
Si 

Hi 

so 
0 

S 

rO 

1 

CO 

a; 

GO 

g 

00" 

as 
0 

■^ 

^ 

--0 

0 

so 

^ 

q 

^j 

;_ 

X 

cc 

5- 

'^ 

00 

0 

1-^ 

-0 

10 

-0 
10 

to 

0 

■30 

S 

^ 

•-0 

S 

S 
©» 

H 

0 
►0 

"* 

CO 

Hi 

,0 

s^ 

<^t 

"--f 

00 

"^ 

«^i 

^ 

^M 

to 

f^ 

«0 

«0 

'~l 

so 

»o 

^ 

0 

bn 

" 

■* 

^ 

■^ 

"^ 

" 

■* 

~* 

" 

■* 

"* 

"* 

~* 

** 

"* 

■* 

"^ 

'-H 

to 

- 

- 

f^ 

- 

- 

- 

- 

- 

Co 

= 

- 

- 

- 

01 

1^ 

»« 

^ 

^ 

:^ 

•* 

;J 

». 

:j 

:: 

^ 

-* 

;^ 

:; 

^ 

:i 

^ 

^ 

CQ 

'--I 


§< 

CQ 


212 


GEADED    ARITHMETIC. 


[Yin.  77 


H 
m 


<5l 


^    -o   ^   sj   Go   ;o   -I 
e^   <st   ci   ?^   '-s   •--(  u> 

t5 

Co 

Co 

So     1^                      ^     "-I     'O 

"-I 

^ 

^ 

So   ^i   Co         Co   e^ 
<to    So    >~i           >o    *-l 

(5t 

tej  ;^ 


«      5^      M 

e   S     • 


.O      K     -tJ 


^ 


■o  .^ 


Q>)     So     VJ-    1-H     C^> 


•S    3 

So     '"H 


<^» 


s^  ^c  -    ::    Co  -    - 


& 
^ 


so 


5^ 


Cj 


so 


S-! 

<to     So 

l^O 

2^    So 

So     ^O 

S'-i 

3.63 

£  1 

oo  «  S  „-  O  *~-  -^ 

se  .-  Qj  J'    o  .-     t 

«   2  ^  §  -=   S   ^ 

Ci  --S  _>;  -o   s^  -ii   -o 

o  -s  ^  -^  ~-  'w  "2 

,o    So    ^  So  "^    "2}    <S(    '~s 

pq 


S!5 


0 

o 
pq 

^  :: 

^  :: 

'-I 

»5 

< 

Co 

so 

•-I 

>~( 

&- 


<X3 


VIII.  78]  teachers'  manual.  213 

78  The  items  on  the  debit  side  of  the  cash  account  are  $137.50, 
$2.40,  $3.70,  $2,  $12.75,  $r)2.50,  $1.96,  $3.60,  $1.26,  $27.90, 
$3.72,  $6.60,  $.54,  $1.15,  and  $3.  The  items  on  the  credit  side 
are  $13.50,  $3.36,  $6.60,  $7.38,  $5,  $27.62,  and  a  balance  of 
$210.12,  making  a  total  footing  of  $273.58.  The  balance  in  A's 
account  is  $62.28  (debit  side),  and  in  B's  account,  $13.50  (credit 
side).  The  last  questions  may  be  postponed  until  after  other 
accounts  have  been  written. 

79-"  83  In  this  exercise,  and  in  each  of  the  seven  following 
exercises,  it  will  be  found  convenient  to  have  the  pupils  use  a 
sheet  of  letter  size  paper  having  twenty-nine  lines,  the  first  and 
second  pages  to  be  used  as  Day  Book,  and  the  third  and  fourth 
pages  as  Cash  Book  and  Ledger.  It  may  be  well  to  go  over 
the  items  of  this  exercise  and  explain  each  one  not  fully  under- 
stood. Wiien  this  is  done,  let  the  pupils  rule  their  papers  and 
make  out  the  account  in  full  before  comparing  it  with  the  account 
given. 

The  following  are  the  missing  items  of  pages  82  and  83,  given 
in  the  order  omitted  : 

Cash  Dr.,  Jan.  15,  To  A.  Lawrence  on  V^j  $20.00  ;  Jan.  25,  To 
A.  Lawrence,  $6.96.  Casli  Cr.,  Jan.  31,  By  balance,  $51.02. 
Footing  of  Cash  account,  $109.77.  Amos  Lawrence  Di".,  Jan.  12, 
To  mdse.,  $2.90.  Amos  Lawrence  Cr.,  Jan.  15,  By  cash  on  account, 
$20.00  ;  Jan.  25,  By  cash  in  full,  $6.96.  Footing  of  Amos  Law- 
rence's account,  $46.96. 

Charles  Smith  Dr.,  Jan.  13,  To  mdse.,  $1.50  ;  Jan.  18,  To  mdse., 
$4.55.  Charles  Smith  Cr.,  Jan.  17,  By  labor,  $5.25 ;  Jan.  18,  By 
wood,  $13.50  ;  Jan.  31,  By  balance,  $23.09. 

Balance  Sheet  Dr.,  Jan.  31,  To  cash  on  hand,  $51.02  ;  Jan.  31, 
To  C.  Smith,  $23.09.  Balance  Sheet  Cr.,  Jan.  31,  By  H.  Brown, 
$18.00  ;  By  net  capital,  $1456.11. 

Net  capital  Jan.  31,  $1456.11.     Net  capital  Jan.  1,  $1319.71. 

84  On  the  following  four  pages  are  the  Day  Book  and  Ledger 
accounts  from  these  items. 


214 


GRADED    ARITHMETIC. 
Day  Book. 


[VIII.  84 


JVest  Newton,  May  1,  1889. 


C.B. 


Mdse.  on  hand. 
Cash        " 


A.  A.  Evans, 

To  15  gal.  N.  0.  7nolasses, 
"   4  i^-  Formosa  tea, 


Hiram  Carter, 

To  2  Ih.  Yt.  butter, 
"   4  lb.  Persian  dates, 

Cr.   - 


C.B. 


C.B. 


By  cash  on  "/c 


A.  A.  Evans, 
By  cash  un  "/c 


8 


Dr. 


To  2  bu.  potatoes. 


11 


Hiram  Carter, 

To  3  lb.  Rio  coffee, 
"    1  box  sardines, 
"   3  lb.  Smyrna  figs, 
"    10  lb.  B.  H.  sugar, 

Cr.   - 


By  2  days''  labor, 
"   2  loads  gravel, 


Dr. 

.60 

.75 


Dr. 

.25 
.15 


Cr. 


.80 


Dr. 

.25 

.20 
.08 


1.87 
.63 


500 

42 


00 
00 


9 

3 


00 
00 


50 
60 


75 
50 
60 
SO 


74 
26 


542 


00 


12 


00 


10 


00 


00 


60 


2 


65 


00 


VIII.  84] 


TEACHERS     MANUAL. 
Day  Book. 


215 


West  Newton,  May  15,  1889. 


A.  A.  Evans, 

To  2\  lb.  Mocha  coffee, 
"  8  lb.  Rangoon  rice, 
"  4  lb.  Malaga  raisins, 

— 17   — 


Iliram  Carter, 
To  3  lb.  currants, 
"    1  bbl.  St.  Louis  flour, 

.    Cr.  


C.B. 


By  cash  on  Vc 
"   1^  cd.  chestnut  wood, 

18   - 


C.B. 


A.  A.  Evans, 
By  cash  on  Ve 


20 


Iliram  Carter, 
By  2  days'"  labor. 


22 


David  Grant, 

To  2  doz.  bananas, 
"   3  gal.  P.  R.  molasses, 

Cr.   ■ 


By  25  gal.  vinegar. 


28 


C.B. 


Iliram  Carter, 
To  cash  on  Vc 


Dr. 

.32 
.10 
.18 


Dr. 

.15 


5.00 


Cr. 


Cr. 

1.38 


Dr. 

.50 
.65 


.19 


Dr. 


SO 

SO 

72 

45 

6 

75 

7 

15 

00 

7 

50 

22 

20 

o 

1 

00 

1 

95 

4 

12 

32 


20 


50 


00 


76 


95 


75 


50 


216 


GKADED    ARITHMETIC. 


[VIII.  84 


CO 


o 


u 
o 


c.  ::i  ^  c>  Q5  »H 
'C^  Co  <o  <^  lo  vj. 


o 


Ci  !>)  Co  i^ 
'--t  '-~i  Q>  <?( 

■    ■    ■  "^j 


O     "^  ■'-3 

05    sT 

•5^-^ 

--i  ■- ^  aj 

O  >' 

?r^'« 

1.(1 

O  '^     CO 

^C5 

lU 

^ 

B 

^  ic  <::b 

;:t^ 

f^ 

l-H    o-l    (5J 

fO 

Sr^    ^ 

K)-    - 

•*  - 

" 

^>  -   ::  '^  Co  'Ci 


S 


^»OC>(S'2)<2iQ><^to'5)'^-->toCo!3<(~^ 
f^i  ^)  <o  '5>  Ql  '^  'Ci  ^>  '^  ^  'O  Co  ^o  «^  ?^  ^ 

■30 

ID 


!;;    00    ^    op 


s  e 


Si 


c-J 


GO     ^ 


^ 


"-I  to  :i 


I 


?^co?^co'o:;  ::  ::  :i  c^ 

^  T^  c^^  ^i 


§ 

"^ 

^> 

'"H 

^! 

'« 


pq 


Si 


<^} 

'~H 

^ 

>> 

!U 

^ 

^ 

CJ  Co 


©^   >~H 


■^ 


^ 


1  = 


VIII.  84] 


teachers'    IVIANUAL. 


217 


<!5 


o 

(0 

h3 


GO 

> 


§ 

■==>? 
^ 

to 

<2> 

■t-H  e-) 

<2.  C5  ^  Co 
<2>  Q>  '^  ?^ 


o 


e  ■ 


Si 


00  Co 


Q)   <a  ®J   ClO 
=i  CO  <30   C> 

e-f  i-H  <Ji  <2) 

CO 

>-<  »-1  ^) 

Pi 


-a  - 
S 

- 

^-- 

;^ 

^ 

1^  eo 

1^ 

- 

- 

*^ 


CO 


<^ 


O     "O  ^  '5^  ^ 


i~-l  >N  ^  ^ 


<5>  CO  >N  «c» 
CC  C>,  Co  »^ 

IS 

>~l  c>  '--1  <*) 
»-H  *s  eo 

CO 

Co 

to 

E^  V}.»^ 

CO 

s.s 

<u 

o  £ 
<'?.:: 

c 

o 

B 

.o 

c  s 

e 

B 

U     09 

•^^ 

.O 

5~ 


§^ 


e^-  -  - 


Si-    -    - 


f^ 

»~H 

?^ 

Ci 

»~H 

>~H 

>-s 

<^< 

s, 

<D 

tr. 

:^ 

^ 

< 

C>  lO    <^   C^    »N 

'-I  to  Q-i  "^  c^ 

to 

<5.i 

'-(  S.J  e^  <?•>  1--I 

''I 

C-, 

W 
W 

X 

in 
w 

< 


Co 


«^ 


S^  Ci 

0^ 

to 
to 

0(  ©( 

Co  ^ 
to  "c 

'~(  ■^  ^  <^} 


•o- 

S£ 

ts 

< 

►<s  - 

si 

^ 

C  "u" 

o 

^o 

o    ii 

e 

ai                  ,    S 

J^ 

S^      a 

^  2 

•♦o 

^.   .   .   . 

E?^ 

^.    . 

?N.  >-H  ?^  Co   t^, 

*"< :: 

i-H  >~i  *-i 

'^H  '~<  9s(  so 

.   ^^ 

So        so 

1=  =  ^  = 

^1= 

218 


GRADED   ARITHMETIC. 


[VIII.  85 


o 


a 


■^     ^       ?. 


pi 

O 

o 
o 
c3 


o 
o 


11 

•+J 

G 

,^3 

'r-* 

S 

-|J 

8 

a 

:/} 

o 

r-< 

m 

o 

Ui 

ri 

4-> 

Q 

5 

^ 

<D 

CJ 

n 

+J 

"^ 

o 

PI 

iX> 


S)^    2 


c;    o 


i:    ph  CO 


o 


cl 


G 


H 
W 
W 

w 

o 

< 


00 

>^   to 

to 

'S'l 

By  Amos  Lawrence, 
"  net  capital, 

^ 

'-I 

To  mdse.  on  hand, 
"   cash        " 

=2>  :: 

GO 

«0 

ito 

?>. 

<S 

»H 

•^ 

'^ 

<^ 

^ 

to 

e-j 

GO 

to 

lf^ 

•^ 

^ 

'-H 

'-H 

«s. 

tiS. 

3 

^ 

s 

GO 

^ 

!~ 

^ 

i'*' 

w, 

-* 

'< 

^ 


& 

^ 


VIII.  86] 


TEACHERS     MANUAL. 


219 


86  This  account  is  supposed  to  extend  from  INIay  2  to  May  20. 
The  last  items  of  the  tirst  page  of  the  Day  Book  should  appear  as 
follows  : 


C.B. 


q 

4 
4 

George  G.  Gates, 
By  3  European  larch, 
"   2  Wisconsin  willow. 

Cr. 

1.12 
.62 

4 

To  cash  on  "/c 

3 
1 

36 
24 

4 


10 


GO 


00 


On  the  second  page  of  the  Day  Book  there  will  be  the  accounts  of 
Gates,  5  lines ;  Hart,  2  lines  ;  Gates,  4  lines  ;  Eaton,  2  lines ;  Hood, 
4  linos  ;  and  Gates,  4  lines.  The  cash  account  on  the  third  page 
of  the  sheet  will  occupy  22  lines,  and  give  room  on  the  yjage  for 
the  Ledger  accounts  of  Isaac  Hart  and  A.  M.  Eaton.  Cash  is 
credited  with  the  following  amounts  :  $4.50,  $9.66,  f  3,  |2.50, 
$3.04,  $10.48,  $2.52,  $10,  $4,  $1.68,  $1.52,  $30,  $2.16,  $3.90, 
$4.74,  $1.76,  $1.24,  $.58,  and  balance  on  hand  May  20,  $298.43. 
On  the  fourth  page  of  the  sheet  will  appear  the  Ledger  accounts  of 
Horace  Hood,  6  lines,  and  G.  G.  Gates,  9  lines,  and  the  Balance 
Sheet.  The  debit  side  of  the  Balance  Sheet  will  be  :  Mdse.  on  hand, 
$1000  ;  Cash  on  hand,  $298.43  ;  Horace  Hood,  $98.30.  The  credit 
side  will  be:  A.  M.  Eaton,  $59.25;  G.  G.  Gates,  $27.74;  Net 
capital  May  20,  $1309.82,  making  a  total  footing  of  $1396.81. 
The  net  gain  ($139.57)  should  be  noted  as  before. 


220 


GRADED   ARITHMETIC. 


[VIII.  87 


S     be 


o 
o 


PI 

o 

o 

CD 
OS 


ft  :S 


CC 


o 
o 
o 

c3 


^-1 
o 

ID 

bJO 
q=l 


cC 


•I— I 


M     be 


pi 

o 
o 
o 
c3 


o 


f-l 
o 


f-l 


CO 
<D 
-1-2 


> 

-1-3 

t-( 
q=l 

o 

w 
•  I—* 


CD 
i-H 

Q 


to  -I-2 

<D    Pi 

uJ      in 


M 


o     P^ 

fi        (T) 


w       ^       =^ 

r5     5     P^ 


?2 


c3     03 


Oi    I — I 

tn 

c3 


-4-2 

p! 
o 


•rH        -1-2       r-H 


be   S 
PI     '^ 

PI     o 
2     bjo 

H      rTJ       -M 
CO 


o  ^ 


r^ 
^ 

•v 

rS=i 

-^2 

CO 

o; 

Cl) 

02 

o 

o 

a 

a; 

a 

^ 

d 

PhW 

G 


H 

H 

W 

M 

CO 

W 
o 

Hi 


<2>  i;^ 
-<5-  so 

ISO    "-s 

■--1  ^ 

^ 

Chas.  Smith, 
Net  capital. 

CO 

1  = 

=::>  ^  Qi 
e>.  ^  (2> 

t^  Q)   t^i 
e^  i5j 

Cash  on  hand, 
Mdse.     " 
Bills  receivable. 

a> 

■4J 

ro 

ra 

tP 

O) 

r^ 

Jh 

-1-2 

-1-2 

q-i 

O 

X 

(B 

O) 

s  +^ 

•-  o 

d 

(V 

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VIII.  89]  teachers'  ]vianual.  221 

89  After  the  first  four  items  uiuler  tlu;  first  date,  Apr.  1,  the 
following  entries  of  the  J)ay  Book  should  be  made  in  order:  First 
page  :  C,  2  items  ;  A,  3  items  ;  B,  3  items  ;  A,  1  item  ;  C,  1  item. 
Second  page  :  B,  3  items  ;  C,  3  items  ;  A,  2  items  ;  C,  3  items  ;  A, 

I  item  ;  A,  3  items.  In  the  Cash  account,  there  will  be  9  entries 
on  the  debit  side  and  13  entries  on  the  credit  side.  C's  Ledger 
account  will  also  be  on  the  third  page  of  the  sheet. 

Balance  Sheet  Dr.:  Mdse.  on  hand,  $3000.00;  Cash  on  hand, 
$2473.38  ;  Amos  Lawrence  (A),  |280.()0  ;  Bills  Receivable,  $250. 
Cr.:  Charles  Smith  (C),  1761.25;  Net  capital,  $5242.73. 

90  This  account  is  supposed  to  extend  from  March  1  to  March  19. 
The  cash  balance  March  9  is  $113.04,  and  May  19,  $199.00.  The 
Balance  Sheet  is  as  follows:  Dr.:  Cash  on  hand,  $199.60;  Mdse. 
on  hand,  $400  ;  Byron  Waters,  $40.00.  Cr.:  Dennis  Rand,  $4.78  ; 
Net  capital,  $634.82. 

92    Answers:    3  26    56.     4 /±  [r/ X  (« -  1)].      5  50    39    10. 

6  '7  =  -^— 4,or,^^^;  w  =  ^^,  or,-^-^-     7  5.     8  6.      10  108. 
n.  —  1  71  —  1  a  a 

II  s='^^Xn.      12  420.      13  $1.45     $19.50. 

It  would  be  well  to  place  upon  the  board  several  series  of  num- 
bers like  the  following : 

(1)  (2)         (3)  (4)         (5)         (6)  (7)         (8) 

13  5  7  9  11  13  15 

4  8  12  16  20  24  28  32 

30  27  24  21  18  15  12  9 

Lead  the  pupils  to  discover  the  fact  that  in  each  of  the  above 
series  of  numbers  there  is  a  constant  difference  between  the  con- 
secutive terms,  and  to  make  a  definition  like  the  following  :  Arith- 
metical progression  is  a  series  of  numbers  which  increase  o.- 
decrease  by  a  constant  difference. 

..Any  term  of  an  arithmetical  series  may  be  found  by  multiplying" 
the  common  difference  by  the  number  of  terms  which  preceded  it. 


222  GRADED    ARITHMETIC.  [VIII.  93 

The  other  rules  may  be  readily  found  from  the  given  examples. 
To  teach  the  rule  for  finding  the  sum  of  an  arithmetical  series, 
place  upon  the  board  any  series,  as  : 

3         6         9       12       15       18 
and,  directly  below  it,  the  same  series  reversed.     The  two  series 
will  appear  as  follows  : 

3  6  9  12  15  18 
21  18  15  12  9  6 
24  +  24  +  24  +  24  +  24  +  24 

=  twice  the  sum  of  the  series.  Therefore  the  rule  :  The  sum  of 
an  arithmetical  series  is  equal  to  the  product  of  one  half  the  sum 
of  the  first  and  last  terms  multiplied  by  the  number  of  terms. 

93  Ansivers:  6  2430.  7  60f.  8  5.  10  98304  131070. 
11  $25.60  $51.15.  12  $59049  $88573.  13  $179.08.  14  9yr. 
2  da.       15  $10737418.21. 

First  place  upon  the  board  two  or  more  series  of  numbers  in 
geometrical  progression,  as  follows  : 

2  6  18  54 

40  20  10  5 

The  pupils  will  see  that  each  of  the  above  series  of  numbers 
increases  or  diminishes  by  a  constant  ratio.  Geometrical  progres- 
sion, therefore,  is  a  series  of  numbers  which  increase  or  diminish 
by  a  constant  ratio. 

To  show  how  any  term  or  ratio  of  a  geometrical  series  may  be 
found,  write  the  factors  of  each  term  of  the  series  upon  the  board ; 
thus  : 

2  6  18  54 

2  2X3  2X3X3  2x3X3X3 

By  questioning  tlie  pupils  upon  the  above  numbers,  tlie  following 
rules  may  be  develo])ed  : 

The  last  term  of  a  geometrical  series  is  equal  to  the  first  term 
multiplied  by  the  ratio  raised  to  a  power  whose  degree  is  one  less 
than  the  number  of  terms.     The  first  term  is  equal  to  the  last 


VIII.  94]  teachers'  manual.  223 

term  divided  by  the  ratio  raised  to  a  power  whose  degree  is  one 
less  than  the  number  of  terms. 

The  ratio  is  equal  to  the  root  whose  index  is  one  less  than  the 
number  of  terms,  of  the  quotient  of  the  last  term  divided  by  the  first 
term. 

The  sum  of  the  series  may  be  found  as  indicated  in  9.  Other 
formulas  may  be  expressed  as  follows  : 

94  Jnsivers:   1  6f/.       2  41 /^Z.       3  2  lb.  @  9/,  1  lb.  @  6/ 

4  ]b.  @  6/,  4  lb.  @  9/.      4  1  lb.  @  50/,  1  lb.  at  65^,  2^  lb.  @  $1. 

5  15  lb.  6  20  lb.,  20  lb.,  GO  lb.  7  12^  gal.  8  1  lb.  2|  oz. 
9  1  lb.  11  oz.  11  pwt.  5.28  gr.  10  3.98  s.  11  G  oz.  12  3  oz. 
5  pwt.       13  $2.45.       14  31%.       15  23.52  oz.     49  lb. 

Many  of  the  questions  and  answers  given  on  the  remaining  pages 
were  given  by  business  men,  mechanics,  or  specialists. 

95  Answers:  1  $39110.  2  $3600  loss.  4  I3G1.07. 
5  $545.43.  6  $216.01.  7  85  lb.  89  lb.  8  Jan.  15.  9  About 
4^%.       10  5%  stock  1%  better. 

96  Answers:  1  $116.68.  2  $172.72.  3  $392.20.  4  $112^ 
$75  5.347%.  5  $8.22.  6  $26.44.  7  29531  bricks. 
8  5276.97984  ft.       9  $21.56.       10  $.128  +  . 

97  Aiiswers:  1  $5,472.  2  Rivets,  f  lb;  burrs,  ^  lb.  3  650.4 
yd.  4  $172.40.  5  $850.  6  5hi%  gain  3^^%  loss.  7  $11.12 
18  rolls     14  yd. 

98  Answers:  1  (a)  8200  ft.;  (h)  7520  ft.;  (c)  4700  ft.;  (d)  151 
bundles;  (e)  108  bundles  ;  (/)  660  ft.;  (y)  $591.39.  2  $87936.51. 
3  A,  $9000  B,  $10125  C,  $7875.  4  $811.69.  5  125  bu. 
140  bu.  6  (a)  374  yd.;  (b)  1  A.  144  rd.;  (c)  80  sq.  rd.;  (d)  108 
sq.  rd. ;   (e)  116  sq.  rd. 

99  Answers:  1  2^9^  cd.  If  cd.  2  105°  15'.  11  720  pass- 
books. 12  Mercury,  5f  as  large  as  the  moon.  14  1350  sq.  in. 
1633^  sq.  in.       15  8.944+  rd.     8  rd.  15.576+  ft. 


224  GRADED    ARITHMETIC.  [VIII.   lOO 

100  Answers :  1  66  rd.  5  ft.  3  A.  5  sq.  rd.  12  sq.  yd.  7  sq.  ft. 
72  sq.  in.  2  $408.41.  3  $44.12.  4  1051  bricks.  5  75.2 
65.2  32.2  31.2  27.2  16.22  15.01  11.94.  6  41  strips  13 
rolls     $2.44     37  yd.       7  646.5+  bbl. 

101  Answers:    1  $10150.       2  253  sq.  ft.  18  sq.  in.       3  $120. 

4  4096  bullets.     5  A,  $512.82   B,  $687.18.     6  20  yd.     7  July  16. 

8  2^s    pi'    pgy    ptt     ara.        9  11^-%   gain     11^%  loss     8%  loss 
15.1%  gain     42f%  gain.       10  41.4+ bbl.       11  56.57+ gal. 

103  Answers:  3  57+%   28+%    68+%.     4  KY.,  1123385932. 

5  S.  Dakota,  1048+%.     6  1880:  16983034673;  1890:  25265581794. 

103  Ansivers:  1  IS  h.  37  niin.  10  sec.  2  5§||aA.  3  4|fi 
Eng.  mi.  4  4071  mi.  5  Wheat,  530|  bu.;  apples,  442  bu.; 
beets,  442  bu. ;  carrots,  442  bu. ;  potatoes,  442  bu. ;  corn,  221  bu. ; 
salt,  2304  bu.  6  80  perches.  7  567 J^  6171  T.  .  8  Wheat, 
33^  bu. ;  rye,  35|  bu. ;  oats,  57^  bu. ;  barley,  41|  bu. ;  salt,  40  bu. ; 
potatoes,  33-J  bu. ;  coal,  25  bu.  9  107Lbu.  10  6^  lb.  11  $26.67. 
12  $10  gain.       13  600  lb.       14  $13.14.       15  Radius,  120  in. 

In  5,  the  approximate  answers  are  given,  the  bushel  being  reck- 
oned as  containing  1^  cu.  ft.;  heaped  bushel,  1^  cu.  ft.,  and  2  bu. 
ears  as  equal  in  bulk  to  1  bu.  shelled  corn. 

104  Ansivers:  1  Pop.  Sweden,  4784675  ;  pop.  U.S.,  62622250  ; 
children,  Belgium,  827928  ;  rate  p.c,  Bavaria,  21.2  ;  rate  p.c.  Neth., 
14.4.       2  £3906  5.S.       3  $500.89.       4  About  13  m.       5  18+%. 

6  44+%.       7  $1600.       8  4  h.  6.54+ min.  P.M.       9  50%     5%. 

105  Answers:  1  $504.07.  2  2d  is  greater  discount  by  2.85%; 
$22.80  saved.  3  $9.75  gain.  4  $436^=*^  $163i7t--  5  59.48+ 
gaL       6  1650  mi.       7  10,;;  h.       8  2096.11     $1855.59     $2511.22. 

9  83^  ft.       10  15(;l%  premium. 

100  Ansivers:  1  33  min.  27.5+ sec.  15  min.  40.1+ sec.  33 
min.  16.8+ sec.  2  78.5+ bu.  133.5+ bu.  307.9+ bu.  3  1302.3+ 
bu.  4  890.1+  gal.  5  1692.02  gal.  35.7+  in.  6  91.19+  bbl. 
9  .019+'"'".       10  1.655012™'="'     1.04+%. 


VIII.  107]  teachers'  manual.  225 

107  Ajiswers;  1  .49  3.94%.  2  3045  meters.  3  59.79 
times.  4  47.31  in.  9.775  in.  5  ().316  sec.  6  144.72  ft. 
305.52  ft.     466.32  ft.       7  3618  ft.       8  188.94  ft.  per  sec. 

108  Ansa-ers :  1  4  ft.  2  11^  in.  from  end  wliere  the  Kilo- 
gram weight  is.  3  6|-  in.  from  middle  on  same  side  as  2  and  3. 
4  129.16^.  5  8  lb.  on  balance  near  the  weight,  and  4  lb.  on  the 
other.  6  11  lb.  on  balance  near  the  12  lb.  weight ;  10  lb.  on  other 
balance. 

Many  solutions  are  possible  for  7;  e.g.  60  lb.  in  middle,  and 
40  lb.  2^  ft.  from  man  ;  or,  40  lb.  in  middle,  and  60  lb.  3-^  ft.  from 
man. 

109  Answers:  1  49+  lb.,  i.e.  anything  in  excess  of  49  lb. 
2  Anything  in  excess  of  1  of  person's  weight.  3  25+ lb.  4  20  + 
lb.     5  73.5+  lb.     6  18849600  lb.     2282568.75  lb.     7  -^\     ^^    i. 

110  Answers:  1  601b.  2  58.92  1b.  3  .84  1b.  4  12  boys 
12  boys.  5  BoxAvood,  .537+;  maple,  .463+.  6  7.29.  7  1.125. 
8  2.56     2.08""".       9  1.13+ .       10  .0000™^.       11  26.32  ft. 

111  Answers:  2  .565+.  3  .376.  4  .307.  5  264  297 
330  352  396  440  495  528.  6  About  43°.  7  About  20 
millions  of  millions  of  miles.       8  12.7+ . 

The  computed  result  of  1  is  18.99+*^™.  The  difference  may  be 
disregarded,  as  it  is  not  greater  than  probable  error  in  reading 
scale  used. 

In  8,  sound  is  reckoned  as  traveling  1120  ft.  per  second. 

113  Answers:  1  A  little  more  than  116°.  2  20.78  in.  3  3.144 
ohms.  4  100.61  ohms  40.24  ohms.  5  -J-  its  weight  at  surface 
^  its  weight  at  surface  0.  6  2800  mi.  from  centre.  7  25  lb. 
11^  lb.     4  lb.     llj  lb.     6i  lb.       8  24000  mi. 

113  Answers:  1  20.8+  sec.  2  Ifi  times  as  loud.  3  4.99+ 
qt.  4  12.58+  pk.  5  34.37+  ft.  6  2500  lb.  1.55+  ft.  7  1  ft. 
1.1+ in.  8  About  117  bbl.  8  2160°  F.  10  2880°  F.  11787.92 
ft.  12  3.61+ sec.  7.82+ sec.  13  9.775  in.  156.4  in.  14  2.9+ 
times  a  second. 


ADVERTISEMENTS 


16 


HILL'S    LESSONS    IN    GEOMETRY. 


•' 


HILL'S  LESSONS  IN  GEOMETRY.     For  the  Use  of  Beginners.     By  G.  A.  Hill, 
A.M..,T^\\i\\oxoii\\eGeo7nc•t^yforIyc'ffil!l!t:rs.     Illustrated.     i2mo.     Cloth.     190  pages. 

For  introduction,  70  cents.     Aiiswcrs,  in  pamphlet  form,  can  be  had  by  teachers. 

In  France  and  Germany,  says  the  President  of  Harvard  Univer- 
sity, they  begin  geometry  at  ten  years  of  age,  and  in  its  right  place, 
namely,  in  connection  with  drawing.  Geometry,  pursued  in  the 
proper  way,  ought  to  be  introduced  in  grammar  grades,  and  within 
a  few  years  it  doubtless  will  be.  The  right  method  is  known,  and 
is  embodied,  it  is  believed,  in  this  book.  The  subject  is  presented 
objectively  and  practically,  instead  of  abstractly  and  theoretically. 
The  central  purpose  is  intellectual  training,  —  teaching  by  practice 
how  to  think  correctly  and  continuously,  —  but  at  the  same  time  the 
main  facts  and  principles  of  geometry  are  taught,  and  also  a  great 
deal  of  the  most  useful  knowledge.  The  exercises,  involving  the  use 
of  simple  instruments  and  drawing  to  scale,  are  of  great  interest  and 
educational  value. 

^EOMETRY  taught  in  this  way  is  interesting,  disciplinary 
and  practically  valuable  in  a  very  high  degree. 


Gives  real  and  living  knowledge. 

For  giving  to  students  who  have  never 
studied  geometry  a  real  and  living  knowl- 
edge of  the  subject  and  a  command  of  its 
more  important  applications,  I  know  of  no 
book  equal  to  this.  —  C.  C.  Rounds, /"/-/«. 
State  A'orDial  School,  Plymouth,  N.H. 


Full  of  inspiration. 

It  is  a  delightful  little  work,  and  full  of 
inspiration.  A  teacher  who  gets  the  spirit 
of  that  book  into  him  cannot  fail  to  teach 
well.  In  the  hands  of  the  pupil  I  know  of 
nothing  that  approaches  it.  —  Corwin  F. 
Palmer,  Si<pt.  of  Schools,  Dresden,  Ohio. 


HILL'S  DRAWING  CASE.  Prepared  expressly  to  accompany  Hill's  lessons  in  Geome- 
try, s.x\d  containing,  in  a  neat  wooden  box,  a  seven-inch  rule  with  a  scale  of  millimeters; 
pencil  compasses,  \\\\.h.  pencil  and  rubber;  ^triangle;  zxid  &  protractor.  Retail  price, 
40  cents;    for  introduction,  30  cents. 

A  specimen  copy  of  the  Lessons  in  Geometry,  with  the  Drawing  Case,  will  be  sent,  post- 
paid, to  any  teacher  on  receipt  of  ^i.oo. 

FRACTIONS.  A  Teachers'  Manual  of  Objective  and  Oral  Work.  By  Helkn  F.  Page, 
State  Normal-Training  School,  Willimantic,  Conn.  8vo.  Boards,  iv  +  47  pages. 
Introduction  price,  30  cents. 

Pupils'  Edition.     Containing  over  three  hundred  e.\amples,  illustrated  with  color- 
diagrams.     Svo.     Boards.     52  pages.     Introduction  price,  30  cents. 

PRIMARY  NUMBER  CARDS.  Prepared  by  Miss  Isabel  Shove,  of  the  George 
Putnam  School,  Boston.  Printed  on  cardboard,  and  boxed  in  sets  of  60.  Price, 
25  cents. 


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WEIiTWORTH'5  ARITHMETICS 

Tljcir    rr;oUo   ')3  rp-a^sVery,  tl^eir  ry)e[])od 

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MONTGOMERY'S  4MERIC4N  HISTORY 

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CLASSICS  FOR  CHILDREN 

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THE  MEW  mTIOMAL  MU5IC  COURSE 
Studied    by  n^orc  pupils  \\)^y)  a.11  other 
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